Journal of Mathematical Sciences, Vol. 172, No. 4, 2011
ON THE SPECTRUM OF THE ALGEBRA OF SINGULARINTEGRAL OPERATORS WITH DISCONTINUITIES INSYMBOLS IN MOMENTA AND COORDINATES
V. Kasatkin
St. Petersburg State University28, Universitetskii pr., Petrodvorets, St. Petersburg 198504, Russia
[email protected] UDC 517.9
We study the C∗-algebra B generated in L2(R) by operators of multiplication by func-
tions with finitely many discontinuities of the first kind and by convolution operators
with the Fourier transforms of such functions. The algebra B is represented as the
restricted direct sum A1 ⊕C A2. We express the spectrum of the restricted direct sum
in terms of the spectra of its summands. This result is used to express the spectrum
of the algebra B in terms of the spectra of A1 and A2. We describe all equivalence
classes of irreducible representations of the algebra B, the topology on the spectrum of
this algebra, and solving composition series. We discuss the abstract index group of the
quotient algebra B by the ideal of compact operators and by the ideal comB generated
by the commutators of elements of the algebra B. Bibliography: 14 titles.
1 Introduction. Results
We denote by B(H) the C∗-algebra of bounded linear operators in a Hilbert space H and
by K(H) the ideal of compact operators. We set K = K(L2(R)). Let R be the one-point
compactification of the real axis, and let X be an arbitrary subset of R. We consider the C∗-subalgebra ΠXC(R) of B(L2(R)) consisting of the operators of multiplication by c that have
the left and right limits c(x−) and c(x+) at each point x ∈ X and are continuous at each point
x ∈ R \X. For such a function the set of its discontinuities (i.e., points, where c(x−) �= c(x+))
is at most countable. Let F be the Fourier transform:
Fu(x) =1√2π
∫
R
e−ixyu(y)dy. (1.1)
Let X, X ⊂ R. The goal of this paper is to study the algebra B(X, X) generated in B(L2(R))
by the subalgebra ΠXC(R) and the subalgebra
F∗(ΠXC(R))def= {FsF−1 | s ∈ ΠXC(R)}. (1.2)
Translated from Problems in Mathematical Analysis 53, January 2011, pp. 49–94.
1072-3374/11/1724-0477 c© 2011 Springer Science+Business Media, Inc.
477
We recall that the spectrum of a C∗-algebra is the set of equivalence classes of its irreducible
representations, equipped with the Jacobson topology. Following [1], we say that a C∗-algebraB is solvable if there exists a composition series
{0} = J−1 ⊂ J0 ⊂ · · · ⊂ Jl = B, (1.3)
consisting of ideals Jj , j = −1, . . . , l, such that there exists an isomorphism
Jj/Jj−1 � C0(Ωj)⊗K(Hj) (1.4)
for j = 0, . . . , l, where Ωj are locally compact Hausdorff spaces. Moreover, the series (1.3) is
said to be solving and the number l is called its length. A solving series of an algebra B is said
to be minimal if B has no solving series of less length.
An operator a ∈ B(H) is called a Fredholm operator if its kernel ker a is finite-dimensional,
the image is closed, and the orthogonal complement coker a of this image has finite dimension.
An operator a ∈ B(H) is a Fredholm operator if and only if its equivalence class is invertible in
B(H)/K(H). The Fredholm index of a Fredholm operator is the integer
ind a = dimker a− dim coker a.
The equality
ind(a+K(H)) = ind a
defines the Fredholm index on the set Inv(B(H)/K(H)) of invertible elements of the quotient
algebra which is constant on its connected components.
By the abstract index group of a unital C∗-algebra B we mean the group Λ(B) of connected
components of the group Inv(B) of invertible elements of the algebra B. If B is a subalgebra
of B(H)/K(H), then there exists a homomorphism Λ(B) → Z associating with the Fredholm
index an element of the abstract index group.
The algebra A (X) is simpler than B(X, X) and is generated in B(L2(R)) by the algebra
ΠXC(R) and the Cauchy integral
(Su)(x) =1
πi
∫
R
u(y)dy
y − x. (1.5)
The algebra A (X) and similar algebras were studied in [2, 3].
In [4, Section 7], the family of operators of algebra B(R,R) of the form
n∑j=1
cjF−1sjF,
where cj , sj ∈ ΠRC(R), was treated. In particular, a matrix-valued function, called the symbol,
was introduced. In terms of this function, with the help of the tools of localizing classes, a
criterion and a formula for index were obtained in [5]. In fact, the symbols introduced in [4]
realize irreducible representations of the algebra B(R, R)/K , although they were not interpreted
in such a way. We also note that, in [4], the operators in Lp(R), p ∈ (1,∞), were considered.
In [6], the algebra B(R, R)/K was studied by means of C∗-algebras. In particular, the ideals
Ix,y of the algebra B(R, R) generated by c ∈ C(R) such that c(x) = 0 and also operators of the
478
form F−1sF , where s ∈ C(R), s(y) = 0, were introduced. The quotient algebras B(R, R)/Ix,y
were described in [6]. On the basis of the results obtained, the essential spectrum of elements
of the algebra B(R, R) was described there. Furthermore, a criterion was obtained and the
Fredholm index was computed for operators of algebra B(R, R) of the form
Op(a)f(x) =1√2π
∫
R
a(x, y)eixyFf(y) dy
satisfying some additional condition. Although the considerations in [6] do not concern the list
of all irreducible representations of the algebra B(R, R), however, in fact, such a list is presented
there.
In this paper, we find all irreducible representations of the algebra B(X, X) (in terms slightly
different from those used in [4, 6]), describe the Jacobson topology on the set of these repre-
sentations, find the minimal solving series, and describe the algebra of symbols, isomorphic to
B(X, X)/K , and its abstract index group, and obtain a formula for the Fredholm index of
elements of B(X, X).
In this paper, we propose an approach based on ideas quite different from the ideas used
in [4, 6]. We use the localization principle for computing the (already known) spectrum of the
algebra A (X), whereas to study the algebra B(X, X), we use, instead of localizing classes as
in [4] and the localization principle as in [6], the formula
B(X, X)/K � A1 ⊕Cn A2,
where the right-hand side contains the restricted direct sum of algebras A1 and A2. The number
n ∈ {1, 2, 4} and the algebras A1 and A2 depend on the sets X and X, but, in all cases, the
structure of these algebras is simpler than that of the algebra B(X, X)/K . For arbitrary
C∗-algebras A1 and A2 the restricted direct sum
A1 ⊕C A2 = {(a1, a2) | a1 ∈ A1, a2 ∈ A2, p1(a1) = p2(a2)}is defined provided that a C∗-algebra C and a homomorphism p1,2 : A1,2 → C are given. The
notion of the restricted direct sum was systematically used in [7], but its spectrum was not
considered there. We show that the spectrum of the restricted direct sum is described by the
formula
(A1 ⊕C A2) = A1 � CA2. (1.6)
We derive this formula in Section 2. Owing to formula (1.6), we can express the spectrum of the
algebra B(X, X)/K in terms of the spectra of the algebras A1 and A2, which can be assumed
to be known.
The author thanks B. A. Plamenevskii for statement of the problem.
1.1 Formulation of the results
We formulate the results in the case X, X � ∞, X �= {∞}. Instead of B(X, X), we write
B. The equivalence class of an element a by an ideal I is denoted by [a]. For x ∈ R we denote
by Ix(X) the ideal of the algebra A (X) generated by c ∈ C(R) such that c(x) = 0. We set
Ifin =⋂x∈R
Ix(X).
479
This ideal and the quotient algebra A /Ifin are studied in Subsection 4.5.
The following assertion, which is the main result of this paper, uses the diagram
B/Kp1 �� ��
p2����
A (X)/Ifin(X)
q1����
A (X)/Ifin(X)q2 �� ��
C4
(1.7)
and the projections p1, p2, q1, q2 given on the generators by the following formulas, where
c ∈ ΠXC(R) and s ∈ ΠXC(R):
p1([c]) = [c], p1([FsF−1]) =
[s(+∞)
1− S
2+ s(−∞)
1 + S
2
], (1.8)
p2([FsF−1]) = [s], p2([c]) =
[c(+∞)
1 + S
2+ c(−∞)
1 − S
2
], (1.9)
q1([S]) = (1, 1,−1,−1), q1([c]) = (c(−∞), c(+∞), c(−∞), c(+∞)), (1.10)
q2([S]) = (−1, 1,−1, 1), q2([s]) = (s(−∞), s(−∞), s(+∞), s(+∞)). (1.11)
Theorem 1.1 (cf. also Theorem 5.10). There exists surjective homomorphisms p1, p2, q1,
q2 satisfying (1.8)–(1.11). The diagram (1.7) is a pullback and defines the isomorphism
B/K � A (X)/Ifin(X) ⊕C4 A (X)/Ifin(X). (1.12)
Suppose that A and B are arbitrary C∗-algebras and r : A→ B is a surjective homomorphism
between them. Denote by r : B → A the corresponding mapping. Its values on representations
π ∈ B are expressed by the equality r(π)(a) = π(r(a)). This mapping is injective, continuous,
and closed.
From Theorems 1.1 and 2.16 it follows that the spectrum(B/K
)is homeomorphic to
(A (X)/Ifin(X)
) �(C4)
(A (X)/Ifin(X)
).
From the description of this homeomorphism we obtain a complete description of the spectrum
of the Jacobson algebra B and the topology on this spectrum. All equivalence classes of nonzero
irreducible representations of the algebra B, except for the class [id] of the identical representa-
tion, can be divided into two sets: the classes p1([π]) for [π] ∈(A (X)/Ifin(X)
)and the classes
p2([π]) for [π] ∈ (A (X)/Ifin(X)). The intersection of these sets consists of four points. The
set U ⊂ B is open in the Jacobson topology if and only if either it is empty or it contains [id]
and its pre-images p−11,2(U) are open. The spectrum of the algebra
(A (X)/Ifin(X)
) � (A (X)) \ (Ifin(X)
)
and the topology on this spectrum are known (cf. Theorems 4.13, 4.16 and Lemma 4.18 below).
By the definition of the restricted direct sum, the algebra B/K is not commutative since
the algebra A (X)/Ifin(X) in (1.12) is not commutative.
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In Proposition 4.20, there is a homomorphism that maps the algebra A (X)/Ifin(X) to the
isomorphic algebra S(A (X)/Ifin(X)
)of matrix-valued functions. Therefore, we can describe
a similar algebra SB that is isomorphic to the algebra B/K . We denote by com(B) the ideal
generated by the commutators [a, b] = ab − ba of elements of the algebra B and by (B)k the
subspace of the spectrum of the algebra B consisting of all k-dimensional representations of this
spectrum. Using the isomorphism
B(X, X)/K � SB,
it is easy to see that the minimal solving series of the algebra B has the form
{0} ⊂ K ⊂ com(B) ⊂ B.
Moreover,
B/ com(B) � C((B)1),
com(B)/K � C0((B)2)⊗K(C2).
We recall that the abstract index group of an arbitrary unital C∗-algebra A is the group
Λ(A) whose elements are connected components of the group InvA of invertible elements of
the algebra A. It is equal to the quotient group Λ(A) = Inv(A)/(Inv(A))0, where (Inv(A))0is the connected component of identity of the group Inv(A). In this case, Λ(B/ comB) is
isomorphic to {0} and Λ(B/K ) is isomorphic to Z. The element of Z corresponding, under the
last isomorphism, to the connected component [b] of an element b ∈ B/K is denoted by ΛInd b.
The Fredholm index is denoted by ind b. On a dense set in Inv(B/K ), the Fredhom index can
be defined by the formula
ind b = ΛInd b =1
2πi
∫
( B)1
d lnU(π) +1
2πi
∫
( B)2
d ln detU(π), (1.13)
where U is the element of the algebra of symbols SB corresponding to b ∈ Inv(B/K ). Since
ΛInd and ind are locally constant, formula (1.13) is sufficient for defining them on the entire
Inv(B/K ).
In the general case (with other X, X) ΛInd b ∈ Zk, k ∈ {0, 1, 2}, it suffices to know ΛInd b
for computing ind b, but the converse assertion is not in general true.
1.2 Structure of the paper
In Section 2, we collect facts concerning restricted direct sums and introduce a mapping of
the spectra spec i and p corresponding to the homomorphisms i and p. In Subsection 2.2, we
prove formula (1.6).
In Subsection 3.1, we study the algebra A0 generated in B(L2(R)) by the operator of multi-
plication by the Heaviside function θ and the Cauchy integral S given by formula (1.5). Using
the Mellin transform, we show that A0 is isomorphic to the algebra S0 of continuous matrix-
valued functions on the axis that have diagonal limits at ±∞. In Subsection 3.2, we introduce
and describe the algebra ΠXC(R) of discontinuous functions on R = R ∪ {∞}. In Subsection
3.3, we formulate sufficient conditions for the compactness of commutators [c, FsF−1].
481
In Section 4, we present some known results concerning the algebra A = A (X) in formu-
lations convenient for our purposes. In Subsection 4.1, we prove that the algebra is irreducible
and contains the ideal K . The localization principle can be applied to the algebra A , which
means that the spectrum of the algebra A can be written in the form
A = {[id]} ∪⋃x∈R
(A /Ix) . (1.14)
The local algebras A /Ix are obtained by the definitions which are not convenient for our
purpose, namely, as the quotient algebras of the algebra A defined by its generators by the
ideal which is also defined by its generators. In Subsection 4.2, we describe a method for
computing such a quotient algebra, i.e., we construct isomorphic quotient algebras admitting
a more convenient description. We complete Subsection 4.2 by applying this method to the
computation of A /Ix. In this case, the method is to choose a family of homomorphisms
ϕn : A → B(L2(R)) such that the mapping ϕ : A → B(L2(R)) : a �→ s-limϕn(a) is defined
and is a homomorphism with kernel Ix. Then the algebra A /Ix is isomorphic to the image
of this homomorphism. For x ∈ X this image is isomorphic to A0, and for x /∈ X the image
coincides with the family of operators of the form α + βS (α, β ∈ C) and is isomorphic to C2.
From (1.14) and the description of the spectrum of local algebras we find the spectrum of the
algebra A /K . The notation is introduced in Subsection 4.3. For any element a of the algebra
A /K we define a function, called the symbol, on the spectrum (A /K ) � π �→ π(a) of the
operator a. We denote by Φ a mapping that with each element of the algebra A /K associates
its symbol. The image of this mapping is denoted by SA and is called the algebra of symbols.
The main result in Subsection 4.3 is Theorem 4.15 which explicitly describes the algebra SA .
Based on this theorem, we describe in Subsection 4.4 the Jacobson topology on the spectrum of
the algebra A /K . In Subsection 4.5, we prove assertions concerning ideals of the algebra A .
These assertions will be useful when we apply the results of Sections 2 and 4 to the algebra B.
In Section 5, we study the algebra B = B(X, X). In Subsection 5.1, we prove that B is
irreducible and contains the ideal K . In the remaining part of Subsection 5.1, we describe the
ideals of the algebra B. In Subsection 5.2, we obtain the formula
B(R, R)/K � A (R)/Ifin(R)⊕C4 A (R)/Ifin(R). (1.15)
Based on this formula, we prove (cf. Theorem 5.10 which is the main result of Subsection 5.3) a
similar formula for B(X, X) with other X and X (cf. formula (5.28)). Subsection 5.4 contains
some consequences of Theorem 5.10: the description of the spectrum of the algebra B/K and
topology there, an isomorphism Φ between B/K and the algebra of symbolsSB whose elements
are functions on the spectrum of the algebra B/K . Finally, we write out the solving series of
the algebra B.
In Section 6, we describe the abstract index group Λ(B/K ) of the algebra B/K . Subsection
6.1 contains the main definitions and general facts concerning the Fredholm index, the abstract
index group, and connections between them. In Subsection 6.2, we prove Lemma 6.7 which is the
main tool for describing Λ(B/K ). The description depends on the pair of sets X, X ⊂ R. All
such pairs are divided into three groups. We compute the isomorphism ΛInd: Λ(SB) → Zk and
derive a formula for the Fredholm index of operators of the algebra B in Subsections 6.3–6.5. In
Subsection 6.6, we show that the abstract index group Λ(B/ comB) for comB �= K consists
of a single element.
482
2 Spectrum of the Restricted Direct Sum of C∗-Algebras
Throughout the section, we assume that all algebras under consideration are C∗-algebras andall homomorphisms between the algebras are *-homomorphisms. We often omit the symbols C∗
and * in the notation.
Definition 2.1. Let A1, A2, C be algebras, and let δ1 : A1 → C and δ2 : A2 → C be
homomorphisms. The restricted direct sum of the algebras A1 and A2 is the algebra
A1 ⊕C A2def= {(a1, a2) ∈ A1 ⊕A2 | δ1(a1) = δ2(a2)}.
In this section, we collect some assertions about restricted direct sums which will be used in
the paper. The main goal of this section is Theorem 2.16 which expresses the spectrum of the
restricted direct sum in terms of the spectra of its summands.
2.1 Pullbacks and Pushouts
In this section, we collect necessary definitions and results of cathegory theory. Then we
formulate Theorem 2.9 and Proposition 2.10 which are reformulations of the corresponding
assertions in [7] adapted for our purposes.
We introduce the notion of a pullback in the cathegory of C∗-algebras whose morphisms
are ∗-homomorphisms and the notion of a pushout in the cathegory of topological spaces whose
morphisms are continuous mappings.
Definition 2.2. A commutative diagram of C∗-algebras
Bβ2 ��
β1
��
A2
δ2��
A1 δ1�� C
(2.1)
is called a pullback if for any algebra D equipped with morphisms γ1 and γ2 such that the
diagram
Dγ2 ��
γ1��
A2
δ2��
A1 δ1�� C
(2.2)
is commutative there exists a unique morphism γ : D → B such that γ1 = β1 ◦γ and γ2 = β2 ◦γ.The algebra B is called a vertex of the pullback (2.1). Pairs of morphisms γ1 and γ2 such that
the diagram (2.2) is commutative are called coherent.
Definition 2.3. A commutative diagram of topological spaces
Y X2f2��
X1
f1
��
Zg1��
g2
��
(2.3)
483
is called a pushout if for any topological space V equipped with morphisms h1 and h2 such that
the diagram
V X2h2��
X1
h1
��
Zg1��
g2
��
(2.4)
is commutative there exists a unique morphism h : Y → V such that h1 = h◦f1 and h2 = h◦f2.The space Y is called a vertex of the pushout (2.3). Pairs of morphisms h1 and h2 such that
the diagram (2.4) is commutative are called coherent.
The assertions about the uniqueness of pullback and pushout are known in cathegory theory.
Proposition 2.4 (uniqueness of a pullback). Let
Bβ2 ��
β1
��
A2
δ2��
A1 δ1�� C
and
B˜β2 ��
˜β1
��
A2
δ2��
A1 δ1�� C
(2.5)
be pullbacks. Then there exists a unique isomorphism σ : B → B such that the following diagram
is commutative:
B
β2
�����������������
β1
��
σ ��������� B
˜β2
��
˜β1
�����������������
A1 δ1�� C A2 .δ2
��
(2.6)
Proposition 2.5 (uniqueness of a pushout). Let
Y X2f2��
X1
f1
��
Zg1��
g2
��
and
Y X2
˜f2��
X1
˜f1
��
Zg1��
g2
��(2.7)
be pushouts. Then there exists a unique isomorphism h : Y → Y such that the following diagram
is commutative:
Y Yh��� � � � � � �
X1
f1
��
˜f1
�����������������Zg1
��g2
�� X2 .
f2
���������������˜f2
��
(2.8)
484
The following assertion describes a pushout vertex.
Proposition 2.6 (isomorphism between a pushout vertex and gluing of topological spaces).
A vertex Y of the pushout (2.3) is isomorphic to the quotient space of the disjoint union X1�X2
by the equivalence relation ∼ generated by {g1(z) ∼ g2(z)}z∈Z .
This quotient space, denoted by X1 �Z X2, is obtained by gluing the topological spaces X1
and X2 along the space Z. We show that an assertion similar to Proposition 2.6 is also valid
for a pullback of C∗-algebras. We consider only the case where the morphisms δ1 and δ2 of the
pullaback 1 are surjective.
Lemma 2.7. We consider a square of the form (2.1). Suppose that the morphisms δ1 and
δ2 are surjective and the following conditions hold:
(1) the morphisms β1 and β2 are surjective,
(2) ker β1 ∩ ker β2 = {0},(3) ker(δ1 ◦ β1) = ker β1 + ker β2.
Then (2.1) is a pullback.
Proof. It suffices to verify that for any C∗-algebra D and any pair of coherent morphisms
γ1 : D → A1 and γ2 : D → A2 there exists a unique mapping γ : D → B such that
γ1 = β1 ◦ γ,γ2 = β2 ◦ γ.
(2.9)
For this purpose, we consider an arbitrary d ∈ D. By condition (1), the homomorphisms β1 and
β2 are surjective. Therefore, there exist b1, b2 ∈ B such that β1(b1) = γ1(d) and β2(b2) = γ2(d).
Let g = δ1 ◦ β1. We have
g(b1) = δ1(β1(b1)) = δ1(γ1(d)) = δ2(γ2(d)) = δ2(β2(b2)) = g(b2).
Hence b2 − b1 ∈ ker g. By (3), there exist b2 ∈ ker β1 and b1 ∈ ker β2 such that
b2 − b1 = b2 − b1. (2.10)
We set b = b1 + b2. Then b = b2 + b1. We have
β1(b) = β1(b1) + β1(b2) = γ1(d) + 0 = γ1(d),
β2(b) = β2(b2) + β2(b1) = γ2(d) + 0 = γ2(d).
Therefore,β1(b) = γ1(d),
β2(b) = γ2(d).(2.11)
Furthermore, for a fixed d ∈ D there exists only one element b ∈ B satisfying relations (2.11).
Indeed, assume that b also satisfies (2.11). Then b − b belongs to the kernel of β1 and to the
kernel of β2 as well. By condition (2), we have b− b = 0.
485
Since there exists a unique element b ∈ B satisfying (2.11), there is a unique mapping
γ : D → B such thatγ1(d) = β1(γ(d)),
γ2(d) = β2(γ(d)).(2.12)
It remains to show that the mapping γ is a *-homomorphism. From (2.12) we find that for
i = 1, 2 and d1, d2 ∈ D
γi(d1 + d2) = γi(d1) + γi(d2) = βi(γ(d1)) + βi(γ(d2)) = βi(γ(d1) + γ(d2)).
Since γ is uniquely defined, we have γ(d1 + d2) = γ(d1)+ γ(d2). Similarly, γ(d1d2) = γ(d1)γ(d2)
and γ(d∗) = (γ(d))∗.
Proposition 2.8. Suppose that the homomorphisms p1 and p2 in the square
A1 ⊕C A2p2 �� ��
p1����
A2
δ2����
A1 δ1�� �� C
(2.13)
are determined by natural projections to the components of the direct sum and the homomor-
phisms δ1 and δ2 are surjective. Then (2.13) is a pullback.
Proof. We show that the diagram (2.13) satisfies conditions (1)–(3) in Lemma 2.7. Let us
check that the mapping p1 is surjective. Consider an arbitrary element a1 ∈ A1. Since δ2 is
surjective, we can find an element a2 ∈ A2 such that δ2(a2) = δ1(a1). Then (a1, a2) ∈ B and
a1 = p1(a1, a2). Since a1 was chosen arbitrarily, p1 is surjective. In a similar way, we can check
that p2 is surjective.
Conditions (2) and (3) follow from the equalities
ker p1 = {(0, a2) | a2 ∈ A2, δ2(a2) = 0},ker p2 = {(a1, 0) | a1 ∈ A1, δ1(a1) = 0},ker(δ1 ◦ p1) = {(a1, a2) | a1 ∈ A1, a2 ∈ A2, δ1(a1) = 0, δ2(a2) = 0}.
The proposition is proved.
From this result and Proposition 2.4 it follows that any pullback (2.1) with surjective δ1 and
δ2 coincides, up to an isomorphism, with the pullback (2.1).
Theorem 2.9. Suppose that the morphisms δ1 and δ2 in the commutative diagram
Bβ2 ��
β1
��
A2
δ2����
A1 δ1�� �� C
(2.14)
are surjective. This diagram is a pullback if and only if the following conditions hold:
486
(1) the morphisms β1 and β2 are surjective,
(2) ker β1 ∩ ker β2 = {0},(3) ker(δ1 ◦ β1) = ker β1 + ker β2.
Proof. By Lemma 2.7, conditions (1)–(3) imply that (2.1) is a pullback.
To check the converse assertion, we consider an arbitrary pullback (2.1) with surjective δ1,
δ2 and the pullback (2.13). From Proposition 2.4 it follows that there exists an isomorphism
σ : B → A such that βi = pi ◦ σ. Conditions (1)–(3) for βi are obtained from similar conditions
for pi, which follow from Proposition 2.8.
The following assertion is used in the proof of the assertion that the square is a pullback (cf.
Subsection 5.2).
Proposition 2.10. The diagram of C∗-algebras
A/(I ∩ J) �� ��
����
A/I
����A/J �� �� A/(I + J) ,
(2.15)
where all the mappings are standard projections, is a pullback for any C∗-algebra A and its closed
ideals I and J .
Proof. We note that
I/(I ∩ J) ∩ J/(I ∩ J) = {0},(I + J)/(I ∩ J) = I/(I ∩ J) + J/(I ∩ J).
Therefore, conditions (1)–(3) in Theorem 2.9 are satisfied. Consequently, (2.15) is a pullback.
2.2 Mappings of spectra
Definition 2.11. Let i : I ↪→ A be an injective homomorphism of C∗-algebras whose image
is an ideal of the C∗-algebra A. If π : I → B(H) is an irreducible representation of the algebra
I, then spec i(π) denotes a unique irreducible representation of the algebra A such that the
following diagram is commutative:
I� � i ��
π��
A
spec i(π)����
����
�
B(H).
We denote by spec i the embedding of spectra I ↪→ A that sends the class [π] of the irreducible
representation π of the algebra I to the class spec i([π])def= [spec i(π)].
487
Definition 2.12. Let p : A � B be a surjective homomorphism of C∗-algebras. If π : B →B(H) is an irreducible representation of the algebra B, then p(π) denotes a unique irreducible
representation of the algebra A such that the following diagram is commutative:
A
p(π) ������
����
�p �� �� B
π��
B(H).
We denote by p the embedding B ↪→ A sending the class [π] of the irreducible representation
π of the algebra B to the class p([π])def= [p(π)].
Remark 2.13. By [8, Subsection 5.5], these mappings are well defined, are continuous, and
are homeomorphisms onto their images. The mapping spec i is open and p is closed. Further-
more, if the sequence
0 �� I� � i �� A
p �� �� B �� 0
is exact, then the images of the mappings spec i and p are complements of each other in A.
Proposition 2.14. Suppose that one of the following diagrams is commutative:
(2.16)
Assume also that the images of i and l (if they are indicated in the diagram) are ideals. Then
the corresponding diagram in (2.17) is defined and is commutative:
(2.17)
Proof. The assertions of the lemma for different diagrams in (2.16) are proved in a similar
way. Therefore, we consider only the third diagram in (2.16). By assumption, the image i(I)
is an ideal of the algebra A. Therefore, the image p(i(I)) is an ideal of the algebra B. On the
other hand,
j(J) = j(q(I)) = p(i(I)).
488
Hence the image of the homomorphism j is an ideal of the algebra B and all the mappings in
the corresponding diagram in (2.17) are defined.
We write ϕ for spec i ◦ p and ψ for q ◦ spec j. The commutativity of the diagrams in (2.17)
follows from the equality ϕ = ψ. We show that the corresponding mappings of irreducible
representations coincide. These mappings are denoted by ϕ and ψ.
Let π : J → B(H) be an irreducible representation of an ideal J , and let u ∈ I. We prove
that
ϕ(π)(i(u)) = ψ(π)(i(u)).
Indeed,
ϕ(π)(i(u)) = spec i(p(π))(i(u)) = p(π)(u) = π(p(u)),
ψ(π)(i(u)) = q(spec j(π))(i(u)) = spec j(π)(q(i(u))) = spec j(π)(j(p(u))) = π(p(u)).
Since u is arbitrary, ϕ(π) and ψ(π) coincide on the image of i.
We use the fact that an irreducible representation of an ideal is extended to an irreducible
representation of the entire algebra in a unique way. In our case, by the definition of spec i,
the restriction of the representation ϕ(π) to i(I) is irreducible and coincides with the restricted
representation ψ(π). Therefore, ϕ(π) = ψ(π). Since π is arbitrary, we have ϕ = ψ.
Theorem 2.15. Let
(2.18)
be a pullback of C∗-algebras. Then
(2.19)
is a pushout.
Proof. We consider the commutative diagram
(2.20)
489
Here, i1,2 are standard embeddings of ker δ1,2 ⊂ A1,2 in A1,2. The embedding j2 : ker δ2 → B
is defined due to the universal property of the pullaback (2.18), regarded as a unique mapping
such that β2 ◦ j2 = i2 and β1 ◦ j2 = 0. Similarly, the equalities β1 ◦ j1 = i1 and β2 ◦ j1 = 0
define the *-homomorphism j1. It is easy to see that the diagram (2.20) is commutative. The
mappings j1 and j2 are injective because the compositions i1 = β1 ◦ j1 and i2 = β2 ◦ j2 are
injective.
It is obvious that the first and third rows of this diagram are short exact sequences. Let
us prove that the second row is also an exact sequence, i.e., j1(ker δ1) = ker β2. The inclusion
j1(ker δ1) ⊂ ker β2 follows from the definition of j1. To verify the inverse inclusion, we define
an homomorphism j : ker β2 → B by the equality j(b) = j1(β1(b)). To verify that it is well
defined, it suffices to note that β1(b) ∈ ker δ1. This relation holds since the right lower square
in the diagram (2.20) is commutative. We show that the homomorphism j coincides with the
identical embedding i of the ideal ker β2 in B. This fact follows from the equalities β1 ◦j = β1 ◦i,β2 ◦ j = 0 = β2 ◦ i and the universal property of a pullback. Thus, we have proved that any
element b ∈ ker β2 is equal to i(b) = j(b) = j1(β1(b)) and, consequently, belongs to the image of
j1. Similarly, all the columns of this diagram are short exact sequences.
We consider the diagram of topological spaces
(2.21)
It is commutative because of the commutativity of (2.20) and Proposition 2.14. According to
Remark 2.13, the images of embeddings β1 and spec j2 do not intersect and their union yields
the entire B. Therefore,
B = β1(A1) ∪ β2(A2).
To show that the right lower square in the diagram (2.21) is a pushout, we consider an
arbitrary topological spaceX and a continuous mapping f1,2 : A1,2 → X such that f1◦δ1 = f2◦δ2.We show that there exists a unique continuous mapping f : B → X such that
f1 = f ◦ β1,
f2 = f ◦ β2.(2.22)
Since
β1(A1) ∪ β2(A2) = B,
the mapping f is uniquely found from (2.22).
Let us show that this notion is well defined. Let b = β1(a1) = β2(a2). By Remark 2.13,
the element a2 belongs either to the image of spec i2 or to the image of δ2. The first variant
is impossible since, in this case, b should belong to the image of spec j2, which contradicts the
490
equality b = β1(a1) in view of Remark 2.13. Thus, a2 = δ2(c) for some c ∈ C. Since the left
lower square in this diagram is commutative, we have β1(a1) = β1(δ1(c)) and, consequently,
a1 = δ1(c). Therefore,
f1(a1) = f1(δ1(c)) = f2(δ2(c)) = f2(a2)
and, consequently, f is well defined.
Let us prove the continuity of f . We choose a closed subset U of X and show that f−1(V )
is closed. We have
f−1(V ) =(f−1(V ) ∩ β1(A1)
)∪(f−1(V ) ∩ β2(A2)
)= β1(f
−11 (V )) ∪ β2(f−12 (V )).
The mapping f1 is continuous. Therefore, the set f−11 (V ) is closed. Since β1 is a homeomorphism
to its closed image, β1(f−11 (V )) is also closed. Similarly, β2(f
−12 (V )) is closed. Therefore, the
union f−1(V ) is also closed.
Theorem 2.15 can be reformulated as follows.
Theorem 2.16. Suppose that A1, A2, and C are algebras and δ1,2 : A1,2 � C are injective
homomorphisms. Then the following canonical isomorphism holds:
(A1 ⊕C A2) � A1 � CA2, (2.23)
defined in the sense of Propositions 2.5 and 2.6 by the pushout
(2.24)
Proof. We consider the diagram
(2.25)
where p1 and p2 are the standard projections. By Proposition 2.8, it is a pullback. By Theorem
2.15, the square (2.24) is a pushout. On the other hand, by Proposition 2.6, the square
(2.26)
491
is also a pushout. By Proposition 2.5, we have the homeomorphism
σ : A1 ⊕C A2 → A1 � C A2. (2.27)
The theorem is proved.
3 Auxiliary Algebras and Compactness of Commutators
3.1 Algebra generated by two projections
Let F be the Fourier transform L2(R) → L2(R) defined on a dense set in L2(R) by the
equality
Fu(p) =1√2π
∫
R
e−ipxu(x)dx. (3.1)
Denote by F∗ the homomorphism of the algebra B(L2(R)) given by the equality
F∗b = FbF−1. (3.2)
In this subsection, we study the algebra A0 generated by the operators θ and S in B(L2(R)),
where θ is the operator of multiplication by a function θ that is equal to 1 for positive values of
variable and vanishes for negative ones, whereas S = −F∗ sgn, wheresgn(x) = 2θ(x)− 1
is the sign function. The operator S can be written in the explicit form:
Su(x) =1
πiv.p.
∫
R
u(y)dy
y − x. (3.3)
For a more convenient description of the algebra, we use the Mellin transform M which acts as
a unitary operator from L2(R) to L2(R)⊕ L2(R) and is given by the formula
Mu(λ) =1√2π
∞∫
0
x−12+iλ
(u(x)
u(−x))dx
or
M = F−1N,
where
Nu(t) = et/2
(u(et)
u(−et)
)
and the Fourier transform is applied in the componentwise sense. LetM∗ be the homomorphism
defined by the equality M∗b = MbM−1. Denote by S0 the image M∗(A0) of the algebra A0.
This algebra is generated by the operators of multiplication M∗(θ) and M∗(S) by the matrices
M∗(θ)(λ) =
(1 0
0 0
),
M∗(S)(λ) =
(− th(πλ) i/ ch(πλ)
−i/ ch(πλ) th(πλ)
).
(3.4)
492
It is easy to verify that the algebra generated by these matrix-valued functions coincides with
the algebra of all continuous functions having diagonal matrix limits as λ→ ±∞. The spectrum
of this algebra consists of points λ in the real axis and four points corresponding to the limits
of diagonal entries of the matrix as λ→ ±∞.
3.2 Algebra of piecewise continuous functions
Let R = R∪{∞} be the one-point compactification of the real axis, and let X be an arbitrary
subset of R. Denote by ΠXC(R) the C∗-algebra generated in L∞(R) by functions continuous on
R, except possibly for finitely many points of the set X where discontinuous of the first kind are
allowed. We identify such functions with operators of multiplication and thereby assume that
ΠXC(R) is a subalgebra of B(L2(R)). For x ∈ R we write ΠxC(R) instead of Π{x}C(R). We
explicitly describe the algebra ΠXC(R) and its spectrum. We begin with the notation.
For c ∈ L∞(R), x ∈ R, we denote by c(x±) the limit limy→x± c(y). Note that for connected
subsets of R the notation 〈a, b〉 is usually used, where a < b and the round or square brackets
should be written instead of the angular brackets. To unify the consideration of connected
subsets of R, we will write 〈a,∞) instead of 〈a,+∞) and (∞, b〉 instead of (−∞, b〉. We also
write the square brackets for ∞; moreover, for a > b we set 〈a, b〉 def= 〈a,∞] ∪ (∞, b〉. Let
I = 〈a, b〉 ⊂ R. We set
ΠXIdef= (I \X) ∪ {x− | x ∈ I ∩X,x �= a} ∪ {x+ | x ∈ I ∩X,x �= b}.
Similarly,
ΠXRdef= (R \X) ∪ {x−, x+ | x ∈ R ∩X}.
We set
ΠX∅def= ∅.
We introduce the topology on ΠX R with the base of sets of ΠXI, where I runs all the intervals,
half-intervals, and segments such that their closed endpoints belong to the set X.
The following theorem describes the algebra ΠXC(R). We note that for X = R the first
assertion immediately follows from [4, Lemma 2.9].
Theorem 3.1.
(1) The set of elements of the algebra ΠXC(R) coincides with the set of functions that are
continuous outside X and possess the left and right limits at each point x ∈ X.
(2) For any c ∈ ΠXC(R) and ε > 0 the inequality |c(x+) − c(x−)| > ε can be valid only for
finitely many points x ∈ X.
(3) The spectrum of the algebra ΠXC(R) is homeomorphic to ΠXR.
Proof. Denote by ΠXC(R) the set mentioned in assertion (1) and verify that this set is a
C∗-algebra. We note that this set is invariant under addition, multiplication, and involution. It
remains to check that it is closed. Suppose that a sequence of functions cn in ΠXC(R) converges
to c. Then there exists limn→∞ cn(y) uniform with respect to y ∈ R. Hence the following limits
exist:
c(x±) = limy→x± c(y) = lim
y→x± limn→∞ cn(y) = lim
n→∞ limy→x± cn(y) = lim
n→∞ cn(x±)
493
for x ∈ X and limy→x
c(y) for x ∈ R \X. Therefore, c ∈ ΠXC(R).
We begin with assertion (2) for c ∈ ΠXC(R). Let ε > 0. The existence of limy→x± c(y) for any
x ∈ R implies the existence of a neighborhood Ix of x such that
|c(y+)− c(y−)| � ε
for all y ∈ Ix \ {x}. In particular, the number of points y ∈ Ix where |c(y+) − c(y−)| > ε is
finite. The sets Ix cover the compact set R. Therefore, the set {y ∈ R | |c(y+)− c(y−)| > ε} is
also finite.
We prove that ΠX R is compact. Let ΠXR be covered by a family of sets {ΠXI}I∈I in the
above-described base of topology. Then for x ∈ X we have
x+ ∈⋃I∈I
ΠXI, x− ∈⋃I∈I
ΠXI,
which implies that some interval containing x is contained in the union of at most two sets in
I. The last assertion is also valid for x ∈ R \X. These intervals cover R and, consequently, we
can extract a finite subcovering. Then the union of ΠXI corresponding to I ∈ I yields a finite
covering of ΠXR.
We note that for any function c ∈ ΠX R its values c(x) are defined at all points x ∈ ΠX R.
Thus, c can be identified with a function ΠXR → C. One can immediately check that the
obtained function is continuous. Taking into account this identification, we have the following
embeddings of C∗-algebras:
ΠXC(R) ⊂ ΠXC(R) ⊂ C(ΠXR).
The first embedding contains the identity and separates points of the space ΠXR. Therefore, by
the Stone–Weierstrass theorem, these algebras coincide. In particular, assertion (1) is proved.
Assertion (2) was checked for c ∈ ΠXC(R) and thereby for c ∈ ΠXC(R). As is known, any com-
pact topological space is homeomorphic to the spectrum of the algebra of continuous functions
defined on this space. Applying this fact to the space ΠX R, we obtain assertion (3).
In the sequel, we identify ΠXC(R) and C(ΠX R).
3.3 Compactness of commutators
We note that, by the inclusion
ΠXC(R) ⊂ B(L2(R)),
the homomorphism F∗ defined by formula (3.2) can be applied to elements s ∈ ΠXC(R).
Theorem 3.2. In each of the following cases, the commutator [c, F∗(s)] is compact:
(1) c ∈ C(R), s ∈ ΠRC(R)
(2) c ∈ ΠRC(R), s ∈ C(R)
(3) c ∈ Π∞C(R), s ∈ Π∞C(R)
The proof of assertions (1) and (2) can be found in [5], and the proof of assertion (3) is
contained in [9].
494
4 Algebra of Singular Integral Operators with Discontinuities
in Coordinates
We fix an arbitrary set X ⊂ R and call it the coordinate set where discontinuity is allowed.
Denote by K the algebra of compact operators in L2(R). This subsection is devoted to the
study of the algebra A generated by functions in ΠXC(R) and the operator S.1) The main
goal of this section is the proof of Theorems 4.13, 4.15, and 4.16. Theorem 4.13 describes the
spectrum of the algebra A /K , Theorem 4.15 deals with the algebra of matrix-valued functions,
isomorphic to the algebra A /K , and Theorem 4.16 concerns the Jacobson topology on the
spectrum (A /K ) .
4.1 Localization in the algebra A
To justify the possibility to apply the localization principle (Theorem 4.7), we first prove
that the algebra A is irreducible. In the following assertions, all the equalities and inclusions
of sets are understood up to a set of zero measure. Denote by χU the characteristic function of
a set U . We use the notation cL = {cl | l ∈ L}, where L is a set of elements l for which the
product cl is defined. For example, if (X, dμ) is a space with measure and U is a measurable
subset of X, then χUL2(X, dμ) is the space of functions in L2(X, dμ) vanishing outside U .
Lemma 4.1. Assume that μ is a σ-finite measure on the set R and Y is a closed subspace
of L2(R, dμ). Then the following assertions are equivalent.
(1) Y = χUL2(R, dμ) for some μ-measurable U ⊂ R.
(2) For all u ∈ Y, v ⊥ Y the supports 2) of u and v do not intersect.
Proof. The implication (1) ⇒ (2) is obvious. Let us verify the implication (2) ⇒ (1).
Assume that assertion (2) is true. We check assertion (1) by constructing a set U . Since μ is
σ-finite, the set R is divided into at most countable number of disjoint sets Rk of finite measure.
To determine the set U , we put
U =⋃k
Uk,∞, Uk,l =
l⋃i=1
(suppuk,i ∩Rk),
where uk,i ∈ Y are found by the following inductive procedure: uk,i is an arbitrary function in
Y such that μ((suppuk,i ∩ Rk) \ Uk,i−1) differs from its maximal value (over all uk,i in Y ) by
at most two times; Uk,0 = ∅. By the above construction, for any k and u ∈ Y the intersection
suppu ∩ Rk belongs to Uk,∞ and, consequently, the support of any u ∈ Y is contained in U ,
which proves Y ⊂ χUL2(R, dμ).
Let Y �= χUL2(R, dμ). Consider an arbitrary nonzero function v ∈ χUL
2(R, dμ) � Y . Since
v ⊥ Y and uk,l ∈ Y , their supports do not intersect. On the other hand, by construction, U
belongs to the union of supports of uk,l, which implies that supp v is contained in the union of
supports of uk,l and, consequently, intersects at least one of them.
1) The definitions of ΠXC(R) and S are given at the beginning of Subsection 3.2 and in (3.3) respectively.2) By the support we mean the set of points where the function is not equal to zero.
495
Proposition 4.2. Any reducing subspace of the algebra C(R) has the form χUL2(R) for
some measurable U ⊂ R.
Proof. Let Y be a reducing subspace of the algebra C(R). By Lemma 4.1, it suffices to
prove that for any u ∈ Y and v ⊥ Y their supports do not intersect. Suppose that u and v are
functions in these spaces such that their supports intersect. Then uv differs from zero and there
is a measurable set V1 ⊂ R such that (χV1u, v) �= 0. Since V1 can be approximated by open sets
as precisely as desired, we can take an open set V2 such that (χV2u, v) �= 0. Using the absolute
continuity of the Lebesgue integral, we take a compact set V3 ⊂ V2 such that
∫
V 2\V 3
|uv| dx < |(χV2u, v)| .
Let c : R → [0, 1] be a continuous function such that c = 1 on V3 and c = 0 outside V2. Then
cu ∈ Y and, consequently, (cu, v) = 0. However, on the other hand,
|(cu, v) − (χV2u, v)| �∫
V 2\V 3
|uv| dx < |(χV2u, v)| .
We arrive at a contradiction.
Proposition 4.3. The algebra A is irreducible.
Proof. We consider a reducing subspace Y of the algebra A . By Proposition 4.2, it has the
form χUL2(R). It remains to prove that the set U coincides with ∅ or R. Let U be nonempty.
Consider an arbitrary interval I intersecting U and a function u that is equal to 1 on I ∩U and
vanishes at the remaining points. By construction, u ∈ Y . Therefore, iπSu ∈ Y . For any x /∈ Ithe integrand in the integral
iπSu(x) =
∫
I∩U
dy
y − x
is of constant sign. Consequently, the integral differs from zero. Hence R \ I ⊂ suppu ⊂ U .
Since I is arbitrary, we have U = R.
Recall that K denotes the ideal of compact operators K(L2(R)) of B(L2(R).
Proposition 4.4. K ⊂ A .
Proof. It is known (cf., for example, [8, Theorem 2.4.9] or [10, Corollary 4.1.10]) that if the
image π(A) : A → B(H) of an irreducible representation of the algebra A contains at least one
nonzero compact operator, then K(H) ⊂ π(A). Thus, we need to check that the algebra Acontains at least one nonzero compact operator. We consider an arbitrary non-constant function
c ∈ C(R). The commutator [c, S] with kernel
1
πi
c(x) − c(y)
y − x
differs from zero. By Theorem 3.2, it is compact.
496
We recall that Propositions 4.3 and 4.4 imply that any nonzero ideal of the algebra Acontains the ideal of compact operators. This fact takes place owing to the following known
assertion (we supply the proof of this assertion for the convenience of the reader).
Proposition 4.5. Suppose that the ideal K(H) of compact operators on the Hilbert space
H is contained in an irreducible algebra B ⊂ B(H). Then K(H) is contained in any nonzero
ideal of the algebra B.
Proof. We consider a nonzero ideal J ⊂ B. As is known, the restriction of any irreducible
representation to the ideal either is equal to zero or is irreducible. The identical representation,
restricted to the ideal J cannot be zero and, consequently, it is irreducible. It suffices to show
that the ideal J contains at least one nonzero compact operator. We consider j ∈ J, j �= 0.
Then there is u ∈ H such that ju �= 0. Denote by Pu the projection onto the space Cu. This
projection is compact and, consequently, belongs to B. The operator jPu differs from zero,
belongs to J , and is compact.
Definition 4.6. We introduce the ideal Ix generated in A by continuous functions in C(R)
vanishing at x ∈ R.
The localization principle was used by Simonenko, Douglas, Dynin, Plamenevskii and Senich-
kin (cf. [11]). In this paper, we use the localization principle in the following form.
Theorem 4.7. Let A be the C∗-subalgebra of the algebra B(H) of bounded operators, and
let C be its commutative subalgebra containing the identity operator. Assume that for all c ∈ C
and a ∈ A the commutator [a, c] is compact. Then
A = {[id]} ∪⋃x∈ C
(A/Jx) .
Lemma 4.8. The spectrum of the algebra A is equal to
{[id]} ∪⋃x∈R
(A /Ix) .
Proof. We apply the localization principle (Theorem 4.7). For the localizing algebra C we
take the algebra C(R). The compactness of the commutator [c, a] for c ∈ C(R), a ∈ A follows
from Theorem 3.2.
Proposition 4.9. For any x ∈ X the isomorphism ϕx : A /Ix → S0 holds, where S0 is the
algebra of matrix-valued functions on the real axis defined in Subsection 3.1. The values of the
isomorphism ϕx on the equivalence classes of the generators of the algebra A take the form
ϕx(c+ Ix) =
(c(x+) 0
0 c(x−)
)for c ∈ ΠXC(R),
ϕx(S + Ix) =
( − th(πλ) i/ ch(πλ)
−i/ ch(πλ) th(πλ)
).
(4.1)
497
For any x /∈ X the isomorphism ϕx : A /Ix → C2 holds. In this case, the values of the
isomorphism on the generators take the form
ϕx(c) = (c(x), c(x)),
ϕx(S) = (−1, 1).(4.2)
This assertion is proved in the following subsection.
4.2 Description of local algebras
The following method is useful for computing local algebras. The idea of the method is
taken from the proof of Proposition 2.1.5 in [12]. Assume that we need to compute the quotient
algebra A/J , where the ideal J (regarded as the closed ideal of the algebra A) is given by the
set of its generator J and the algebra itself (regarded as a Banach algebra) is given by the set
of generators A ∪ J .To find these quotient algebras, we choose a family of homomorphisms ϕn : A→ B(H) and
define a mapping ϕ : A→ B(H) by the equality ϕ(a) = s-limn
ϕn(a) on elements where the limit
exists. Moreover, the mapping ϕ must satisfy the following conditions:
(i) ϕ is defined on J and vanishes on J ,
(ii) ϕ is defined on A,
(iii) for any finite combination p formed from elements of A by addition, multiplication, and
involution the following inequality holds: ϕ(p) � ‖p+ J‖.
As shown in Lemma 4.11, ϕ is a *-homomorphism defined on the entire algebra A and its
kernel coincides with the ideal J . Therefore, the quotient algebra A/J is isomorphic to the
image ϕ(A) generated by ϕ(a) for a ∈ A.
Owing to the following lemma, we can simplify the verification of conditions (i) and (ii)
Lemma 4.10.
(1) Let for every a ∈ J the limits limnϕn(a)u vanish for all u in a dense subset of the Hilbert
space H (such sets can be different for different a). Then condition (i) is satisfied.
(2) Assume that for every a ∈ A the limits limnϕn(a)u exist for all u in a dense subset of the
Hilbert space H (such sets can be different for different a). Then condition (ii) is satisfied.
Proof. Both assertions will be proved simultaneously. Assume that a ∈ J if we prove
assertion (1), and a ∈ A if we prove assertion (2). By condition, limnϕn(a)u exists for all u in
some dense subset U of the space H. We prove that this limit exists on the entire space H. For
this purpose it suffices to show that for all u ∈ H the sequence ϕn(a)u is a Cauchy sequence.
Consider an arbitrary ε > 0 and u0 ∈ U such that
‖a‖ ‖u− u0‖ < ε/3.
498
Since ϕn(a)u0 is a Cauchy sequence, from the above inequality for sufficiently large n and m we
have
‖ϕn(a)u− ϕm(a)u‖ � ‖ϕn(a)u− ϕn(a)u0‖+ ‖ϕn(a)u0 − ϕm(a)u0‖+ ‖ϕm(a)u0 − ϕm(a)u‖� ‖a‖ ‖u− u0‖+ ‖ϕn(a)u0 − ϕm(a)u0‖+ ‖a‖ ‖u0 − u‖< ε/3 + ε/3 + ε/3 = ε.
The existence of s-limn
ϕn(a) follows because B(H) is strongly complete. In the case a ∈ J ,
from the assumptions of the lemma we find that the limit operator vanishes on a dense set and,
consequently, is the zero operator.
Lemma 4.11. If conditions (i)–(iii) are satisfied, then the mapping ϕ is a *-homomorphism;
moreover, kerϕ = J .
Proof. The linearity of ϕ follows from its definition. It suffice to prove the following prop-
erties:
(1) ϕ(ab) = ϕ(a)ϕ(b) for all a, b ∈ Dom(ϕ),
(2) Dom(ϕ) = A,
(3) ϕ(a∗) = (ϕ(a))∗ for all a ∈ Dom(ϕ),
(4) ker(ϕ) = J .
Let us prove property (1). We recall that the product of operators is continuous in the sense
of the strong convergence. Assume that
ϕ(a) = s-limn
ϕn(a) and ϕ(b) = s-limn
ϕn(b).
Then the strong continuity of the product implies the existence of the limit
ϕ(ab) = s-limn
ϕn(ab)
and the equality
ϕ(ab) = ϕ(a)ϕ(b).
Consider property (2). As was already established, Dom(ϕ) is a subalgebra of A. Since
Dom(ϕ) contains A and J generating A (regarded as a Banach algebra), in order to prove
Dom(ϕ) = A it suffices to show that Dom(ϕ) is closed. Let am be a sequence of elements
in Dom(ϕ), and let am → a ∈ A. Then ϕn(am) converges to ϕn(a) as m → ∞. Since the
homomorphisms ϕn do not increase the norms, ϕn(am) converges uniformly with respect to
n. By the continuity of the strong limit with respect to the uniform convergence, the limit
ϕ(a) = s-limn
ϕn(a) exists and is equal to limmϕ(am).
To prove property (3), we check the equality
s-limn
ϕn(a∗) = (s-lim
nϕn(a))
∗.
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By property (2), the left-hand and right-hand sides of this equality are defined. Therefore, it
suffices to check this equality for weak limits. But this assertion is valid since the weak limit
and *-homomorphisms ϕn are invariant under involutions.
Finally, we prove property (4). We note that J ⊂ ker(ϕ) follows from the fact that ker(ϕ)
is a closed ideal containing J . Therefore, the mapping Φ: A/J → B(H) given by the equality
Φ(a+ J) = ϕ(a) is well defined. We show that Φ is injective. It suffices to prove that ‖Φ(a)‖ �‖a‖ on a dense set in A/J . For such a set we take the image under the standard projection
A → A/J of the set of finite combinations described in (iii) since the required inequality is
formulated in this condition.
Proof of Proposition 4.9. We apply the above method to the algebra A . We reduce the
consideration of the case of an arbitrary point x ∈ R to the case x = 0. For this purpose,
let us look how the algebra A changes under the change of variables. Suppose that a mapping
f : R → R is continuous, bijective and has the derivative f ′(x) which is defined, continuous and is
not equal to zero everywhere, except possibly for finitely many points. Then the inverse mapping
f−1 possesses the same properties. We define the corresponding operator Ef : L2(R) → L2(R)
by the equality
Efu(x) =√
|f ′(x)|u(f(x)).It is easy to verify that it is unitary and (Ef )
−1 = Ef−1 . We consider the corresponding
automorphism (Ef )∗ of the algebra B(L2(H)) given by the equality
(Ef )∗(a) = Efa(Ef )−1.
If a is the operator of multiplication by a(x), then (Ef )∗(a) is the operator of multiplication by
a(f(x)).
It is easy to check that if f(x) = (αx+ β)/(γx + δ) and αδ − βγ > 0, then (Ef )∗ sends theoperator S to the operator (Ef )∗(S) = S, the algebra A = A (X) to the algebra (Ef )∗(A (X)) =
A (f(X)), and the ideal Ix = Ix(X) to the ideal (Ef )∗(Ix(X)) = If(x)(f(X)). We note that
for any point x0 ∈ R it is possible to find f such that f(x0) = 0. Therefore, it suffices to prove
the assertion for x = 0.
Now, we compute the ideal A /I0. For J we take the set of functions c ∈ ΠXC(R) vanishing
at 0, i.e., c(0−) = c(0+) = 0. For A we take the operator S for 0 /∈ X and the pair {S, θ} for
0 ∈ X, where θ is the Heaviside function.
We note that an arbitrary function c ∈ ΠXC(R) can be represented as the sum of three
terms (c(0+) − c(0−))θ, c(0−)S2, and a function vanishing at 0. Therefore, A ∪ J generates a
Banach algebra A . Finally, we set ϕn = (Ex �→x/n)∗.Let us check (i)–(iii). We note that (ii) and (iii) are valid because ϕn(S) = S and ϕn(θ) = θ.
To prove (i), we use (4.10) which asserts that it suffices to verify the equality limnϕn(c)u = 0
for elements u in a dense subset of L2(R). For such a subset we take the set of compactly
supported functions. For c ∈ J and any ε > 0 there is an interval containing 0, where c < ε.
Then (ϕn(c)u)(x) = c(x/n)u(x) and the required convergence follows from the fact that for
sufficiently large n the inequality |c(x/n)| < ε holds on the support of u.
Thus, A /I0 is isomorphic to the image of the algebra A under the mapping ϕ. For x /∈ X
the image of the algebra A is generated by the operator S and is isomorphic to C2 since S2 = 1.
For x ∈ X this image is generated by the operators θ and S and, consequently, it coincides with
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the algebra A0 considered in Subsection 3.1. Formulas for the images of basis elements indicated
in the formulation of this assertion are obtained from (3.4).
4.3 Spectrum and the algebra of symbols
Using the isomorphism ϕx in Proposition 4.9, we describe some representations of the algebra
A . Suppose that x ∈ X and a ∈ A . Then ϕx(a + Ix) ∈ S0 is a matrix-valued function. For
λ ∈ R ∪ {−∞,∞} we define the π(x, λ) of the algebra A by the equality
π(x, λ)(a) = ϕx(a+ Ix)(λ).
Entries of the matrix π(x, λ)(a) are denoted by (π(x, λ)(a))α,β , where α, β ∈ {“+”, “−”}. Usingthe equality (4.1) in Proposition 4.9, we find
π(x, λ)(c) =
(c(x+) 0
0 c(x−)
)for c ∈ ΠXC(R), (4.3)
π(x, λ)(S) =M∗(S)(λ) =( − th(πλ) i/ ch(πλ)
−i/ ch(πλ) th(πλ)
). (4.4)
We recall that the matrix-valued function M∗(S) was introduced in Subsection 3.1. Suppose
that x /∈ X and a ∈ A . Then ϕx(a+ Ix) ∈ C2. The components of this vector are denoted by
π±(x)(a). From the equality (4.2) in Proposition 4.9 we find
π±(x)(c) = c(x) for c ∈ ΠXC(R),
π±(x)(S) = ±1.(4.5)
The identity 2×2-matrix is denoted by Id2. The representations π(x, λ) for x /∈ X, λ ∈ R ∪{−∞,+∞} are defined by the formula
π(x, λ)(a) =π+(x)(a) − π−(x)(a)
2M∗(S)(λ) +
π+(x)(a) + π−(x)(a)2
Id2, (4.6)
and the representations π±(x±) for x ∈ X are defined by the formula
πα(xβ)(a) = π(x,−αβ∞)β,β(a), α, β ∈ {“+”, “−”}. (4.7)
In this case, (4.3) and (4.4) hold for all x ∈ R, whereas (4.5) holds for all x ∈ ΠX R.
The ideal K is contained in all the ideals Ix. Therefore, the above representations annihilate
K and can be regarded as representations of the quotient algebras A /K for which we preserve
the notation π(x, λ), π±(x).
Remark 4.12. The above representations introduced for different X agree: if a ∈ A (X1)∩A (X2), then π(x, λ)(a) is independent of the choice of a set X under consideration. It suffices
to verify this assertion for X2 = R: for the remaining X2 it follows by transitivity. For X2 = R
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we have X1 ⊂ X2 and A (X1) ⊂ A (X2). The set X does not occur explicitly in formulas (4.3),
(4.4), and (4.5). Therefore, the restriction of the representation of the algebra A (X2) to the
algebra A (X1) coincides with the representation in A (X1) on the basis elements of the algebra
A (X1). Hence they also coincide on A (X1).
Theorem 4.13. The spectrum of the algebra A /K consists of representations
{π±(x) | x ∈ ΠX R} ∪ {π(x, λ) | x ∈ X, λ ∈ R}.
Proof. The assertion follows from Lemma 4.8, Proposition 4.9, and the description of the
spectrum of the algebra A0 in Subsection 3.1.
In view of the identification of representations of the algebras A and A /K , the first part
in the theorem concerns one-dimensional representations of the algebra A and the second part
concerns two-dimensional ones. They are denoted by (A )1 and (A )2 respectively. By a function
on (A /K ) we mean a function taking the values in C on (A )1 and the values in the set of
2×2-matrices on (A )2. We define a homomorphism Φ that associates with an element a of
the algebra A /K the corresponding function Φ(a) = Φa on (A /K ) given by the equality
Φa(π) = π(a). We set
‖Φa‖ = supπ∈(A /K )
‖Φa(π)‖ .
As is known, in this case, ‖Φa‖ = ‖a‖ and ker Φ = {0}. For the sake of convenience, for a ∈ Awe set Φ(a) = Φ(a + K ). The image under the *-homomorphism Φ is denoted by SA and is
called the algebra of symbols of A .
Lemma 4.14. Let U be a function on (A /K ) such that U ◦ π± ∈ C(ΠX R). Then there
exists a unique element a0 ∈ A of the form α+βS, where α, β ∈ ΠXC(R), such that U −Φ(a0)
vanishes on (A )1. The following equality holds:
a0 =∑±
(U ◦ π±)(1± S)/2.
Proof. Let a0 = α+ βS. By assumption, U −Φ(a0) vanishes on (A )1, i.e.,(U −Φ(a0)
) ◦ π± = 0.
We transform the left-hand side of this equality as follows:(U − Φ(a0)
) ◦ π± = (U ◦ π±)− (x �→ π±(x)(a0)) = (U ◦ π±)− (α± β).
Adding and subtracting the obtained equalities corresponding to different choice of the sign “±,”
we find
α =(U ◦ π+) + (U ◦ π−)
2,
β =(U ◦ π+)− (U ◦ π−)
2.
Thus, α and β are uniquely restored from the conditions on a0, which implies the required
assertion. The formula for a0 follows from the explicit formulas for α and β.
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Theorem 4.15. In the image of SA under the *-homomorphism Φ, there are only those
functions U that satisfy the following conditions.
(1) U ◦ π± ∈ C(ΠX R).
(2) For each x ∈ R the function λ �→ U(π(x, λ)) is continuous (for λ ∈ R) and there exists
limλ→±∞
U(π(x, λ)) =
(U(π∓(x+)) 0
0 U(π±(x−))
). (4.8)
(3) Let a0 ∈ A be the element in Lemma 4.14 corresponding to U . The function (x, λ) �→(U − Φa0)(π(x, λ) defined on the locally compact space X × R, where X is equipped with
the discrete topology and R is equipped with the usual topology on the real axis, converges
to zero at infinity. In other words, for any ε > 0 the following set is finite:
{x ∈ X | sup
λ‖(U − Φa0)(π(x, λ))‖ > ε
}. (4.9)
Proof. 1◦. We show that (1)–(3) hold on SA . The algebra SA is generated by ΦS and Φc
for c ∈ ΠXC(R). From (4.3), (4.4), (4.5) and the equality Φa(π) = π(a) we find
Φc(π±(x)) = c(x), Φc(π(x, λ)) =
(c(x+) 0
0 c(x−)
), (4.10)
ΦS(π±(x)) = ±1, ΦS(π(x, λ)) =M∗(S)(λ) =
( − th(πλ) i/ ch(πλ)
−i/ ch(πλ) th(πλ)
). (4.11)
These formulas show that Φc and ΦS satisfy conditions (1) and (2). These conditions are
preserved under taking the sum, product and involution and the limit passage. Therefore, these
conditions also hold on the entire algebra.
We turn to condition (3). We show that it holds on the ideal J of the algebra SA generated
by the commutators [Φc,ΦS] for c ∈ ΠXC(R). We note that U(π±(x)) = 0 for U ∈ J, x ∈ ΠX R
and, consequently, a0 = 0.
We begin with U = [Φc,ΦS]. Let ε > 0. By Theorem 3.1, the set
{x ∈ X | ‖c(x+)− c(x−)‖ > ε/2}
is finite. If an element x ∈ X does not belong to this set, then
∥∥[Φc,ΦS ](π(x, λ)
)∥∥ =∥∥[(c(x+)− c(x−)
)M∗(θ) + c(x−)Id2,M∗(S)(λ)
]∥∥=∥∥[(c(x+)− c(x−)
)M∗(θ),M∗(S)(λ)
]∥∥� 2 ‖c(x+)− c(x−)‖ ‖M∗(θ)‖ ‖M∗(S)‖ � ε. (4.12)
Thus, condition (3) holds for U = [Φc,ΦS ]. It is clear that it also holds on the entire ideal J .
We prove that any element U of the algebra SA is represented as
U = ΦcΦS +Φs + V, (4.13)
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where c, s ∈ ΠXC(R) and V ∈ J . For this purpose, we consider the quotient algebra SA /J .
Since the generators Φc and ΦS commute modulo J , their product is equivalent modulo J to an
element of the form ΦcΦS or Φs. Therefore, any combination of generators is equivalent to the
sum ΦcΦS +Φs. The representation (4.13) follows from the closedness of such sums.
Now, for arbitrary U ∈ SA we have the representation (4.13). From this representation it
follows that a0 = cS + s and U − Φa0 = V . Therefore, condition (3) for U is obtained from the
same condition (3) for V .
2◦. We show that any function U satisfying (1)–(3) belongs to SA . For this purpose, it
suffices to show that V = U − Φa0 belongs to SA . By (3), for any ε > 0 everywhere, except
possibly for finitely many points xi, we have
supλ
‖V (π(x, λ))‖ < ε.
Therefore, finite sums of functions Vi coinciding with V for x = xi and vanishing for the re-
maining x approximate V as precisely as desired. Thus, it suffices to verify that V ∈ SA for
functions V taking nonzero values only at one value of x = x0 ∈ X. Using again the closeness
of SA , we assume that V (π(x0, ·)) is compactly supported.
Consider a function c0 in Πx0C(R) such that c0(x0+)− c0(x0−) = 1. Let W = [ΦS ,Φc0 ]. It
is easy to check thatW vanishes for x �= x0 and det(W (π(x0, λ))) differs from zero for all λ ∈ R.
By the compactness of support of V (π(x0, ·)), the matrix-valued function
f(λ) = V (π(x0, ·))(W (π(x0, λ)))−1 (4.14)
is continuous and converges to zero at infinity. Therefore, f ∈ A0 and there is an element b ∈ A
such that f = ϕx0(b + Jx0) (here, ϕx0 is the *-homomorphism from Proposition 4.9). But, in
this case, for W1 = Φ(b+ K ) ∈ S0
W1(π(x0, λ)) = f(λ). (4.15)
Let us check that V = W1W. This equality is valid on one-dimensional representations and
on the representations π(x, λ) for x �= x0 since V and W vanish there. On the representations
π(x0, λ), the equality follows from (4.14) and (4.15).
Corollary 4.15.1. Suppose that U ∈ SA and ε > 0. Then for all x ∈ X, except possibly
for a finite number of points,
supλ
∥∥∥∥∥U(π(x, λ)) −∑±U(π±(x+))(1 ±M∗(S)(λ))/2
∥∥∥∥∥ < ε.
Proof. It suffices to consider points x ∈ X that belong to neither the set (4.9) for ε = ε/10
nor the set of points x ∈ X such that |c(x+)− c(x−)| > ε/10 for c = U ◦ π±.
If U ∈ SA , then there is a unique a ∈ A /K such that U = Φa. If π is a representation of
the algebra A /K (not necessarily irreducible), then we set U(π) = π(a). Using this notation,
we can write, for example,
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U(π(x,±∞)) =
(U(π∓(x+)) 0
0 U(π±(x−))
).
If U(π±(x+)) = U(π±(x−)) for x ∈ X, then U(π±(x)) denotes their common value.
4.4 Jacobson topology on the spectrum
We describe the Jacobson topology on the spectrum of the algebra A /K by describing the
fundamental system of neighborhoods of points of this spectrum. We recall that the definition
of ΠXI for intervals, half-intervals, and segments I ⊂ R is given in Subsection 3.2.
Suppose that x ∈ X and J ⊂ R is an interval. We set
Vx0,Jdef= {π(x0, λ) | λ ∈ J} . (4.16)
Let α ∈ {“+”, “−”}, and let I be an interval in R. We set
V αI
def= {πα(x) | x ∈ ΠXI} ∪ {π(x, λ) | x ∈ I ∩X,λ ∈ R} . (4.17)
Suppose that x0 ∈ X, α, β ∈ {“+”, “−”}, I ⊂ R is a half-interval with the closed endpoint x0such that x0β ∈ ΠXI and J ⊂ R is a semi-infinite interval going to αβ∞. We set
V αI,x0β,J
def= {πα(x) | x ∈ ΠXI}∪{π(x, λ) | x ∈ I ∩X \ {x0}, λ ∈ R}∪{π(x0, λ) | λ ∈ J} . (4.18)
Theorem 4.16.
(1) For points π(x0, λ0) with x0 ∈ X, λ0 ∈ R the fundamental system of neighborhoods consists
of the sets Vx0,J with J � λ0.
(2) For points πα(x0) with x0 ∈ R \X, α ∈ {“+”, “−”} the fundamental system of neighbor-
hoods consists of the sets V αI with I � x0.
(3) For points πα(x0β) with x0 ∈ X, α, β ∈ {“+”, “−”} the fundamental system of neighbor-
hoods consists of the sets V αI,x0β,J
.
Proof. We note that the base of the Jacobson topology on (A /K ) is formed by the set
NUdef= {π | U(π) �= 0}, U ∈ SA . (4.19)
Let us check that any interval Vx0,J is open in the Jacobson topology. We consider a con-
tinuous matrix-valued function on the line {π(x0, λ), λ ∈ R} that is different from zero only on
this interval and converges to zero at infinity and extend this function by zero to (A /K ) . By
Theorem 4.15, the function obtained U belongs to SA . Since Vx0,J = NU , the set Vx0,J is open.
By Theorem 4.15, the restriction of any function U ∈ SA to {π(x0, λ), λ ∈ R} is continuous.
Therefore, on any set NU containing the point π(x0, λ0) there is an interval of the form Vx0,J .
Consider neighborhoods of the second kind. Let us check that the V αI is open. We choose
a function c ∈ C(R) such that I = {x ∈ R | c(x) �= 0} and note that V αI = NU for U =
Φ([c(1 + αS)/2]).
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Let x0 �= X, α ∈ {“+”, “−”}. We show that for any U ∈ SA such that U(πα(x0)) �= 0 the
set NU contains V αI for some I � x0. We set ε = |U(πα(x0))| /3. If I ⊂ R is a sufficiently small
neighborhood of x0, then for x ∈ ΠXI
|U(πα(x))| > 2ε.
By Corollary 4.15.1, decreasing the neighborhood I if necessary, for x ∈ I ∩X,λ ∈ R we have
‖U(π(x, λ))‖ > −ε+∥∥∥∥U(π+(x))
1 + S
2+ U(π−(x))
1 − S
2
∥∥∥∥= −ε+max
(U(π+(x)), U(π−(x))
)
> −ε+ 2ε > 0. (4.20)
Finally, we consider neighborhoods of the third kind. Let us check that the set V αI,x0β,J
is
open. We choose a function c ∈ Πx0C(R) such that the equality c(x) = 0 holds for those and
only those x ∈ ΠX R for which x /∈ ΠXJ . We set U1 = Φ([c(1 + αS)/2]). Let U be a function
that can differ from U1 only on the line {π(x0, λ) | λ ∈ R} and is continuous on this line;
moreover, it has the same limits as U1, as λ → ±∞, and is different from the zero matrix only
if λ ∈ J . Such a function exists since the set J is open and contains a neighborhood of infinity
where limλ→±∞
U1(π(x0, λ)) is not equal to zero. Under the passage from U1 to U , none of the
assumptions of Theorem 4.15 fails. Therefore, U ∈ SA . By construction, V αI,x0β,J
= NU .
If U is different from zero at πα(x0β), the set NU contains a set of the form V αI,x0β,J
. This
assertion is proved in the same way as a similar assertion for V αI .
4.5 Quotient algebra A /Ifin
We introduce the ideal Ifin of the algebra A by the equality
Ifin =⋂x∈R
Ix. (4.21)
There is a one-to-one correspondence between the set of ideals of an algebra and the set of
closed subsets of its spectrum. The ideal corresponding to such a subset consists of operators
that are sent to zero by all representations in this subset. The proof of these assertions can be
found in [8, Section 5.4]. Since any representation of the algebra SA is obtained by computing a
function at a point, all its ideals have the form {U ∈ SA | U |S = 0}, where S is a closed subset
of (A /K ) . For the sake of simplicity, we identify representations of the algebra SA with the
corresponding points.
Lemma 4.17. The following assertions hold.
(1) For x /∈ X the ideal Φ(Ix) coincides with the set of functions U ∈ SA vanishing at π±(x).
(2) For x ∈ X the ideal Φ(Ix) coincides with the set of functions U ∈ SA vanishing at the
points πα(xβ) with α, β ∈ {“+”, “−”} and at the points π(x, λ) with λ ∈ R.
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Proof. By the definition of the ideal Ix, the set of points described in the lemma coin-
cides with the set of representations vanishing on Ix. Taking into account this correspondence
between ideals and closed subsets of the spectrum, we arrive at the assertion of the lemma.
Lemma 4.18. The following assertions hold.
(1) If ∞ /∈ X, then the ideal Φ(Ifin) coincides with {0}.(2) If ∞ ∈ X, then the ideal Φ(Ifin) coincides with the ideal of functions of the algebra SA
that vanish outside the set {π(∞, λ) | λ ∈ R}.
Proof. The assertions immediately follows from the definition of the ideal Ifin, Lemma 4.17,
and the following assertion which can be directly derived from [8, Section 5.4]. Suppose that
Iα is a family of ideals of some C∗-algebra and Sα is the closed subset of the spectrum of this
algebra associated with Iα. Then the intersection of ideals Iα corresponds to the closure of the
union of sets Sα.
Lemma 4.19. Let ∞ ∈ X, and let c be an arbitrary function that is discontinuous at
infinity and is continuous on the real axis. Then the ideal Ifin of the algebra A is generated by
the commutator [c, S].
Proof. If Φ is regarded as a *-homomorphism from A into SA , then kerΦ = K . By
Proposition 4.5, the ideal K is contained in any nonzero ideal of the algebra A . Therefore, it
suffices to prove that the ideal Φ(Ifin) is generated by Φ([c, S]). By Lemma 4.18, it suffices to
verify that the set where Φ([c, S]) is different from zero coincides with the set {π(∞, λ) | λ ∈ R}.It is obvious that Φ([c, S]) = [Φ(c),Φ(S)] vanishes at the points (A )1. For x ∈ X \ {∞} the
matrix Φ(c)(π(x, λ)) is proportional to the identity matrix and, consequently, commutes with
Φ(S)(π(x, λ)). For x = ∞, λ ∈ R we have
Φ ([c, S])(π(x, λ)
)= π(x, λ) ([c, S]) =
[(c(x+) 0
0 c(x−)
),
( − th(πλ) i/ ch(πλ)
−i/ ch(πλ) th(πλ)
)]
=(c(x+)− c(x−)
) [(1 0
0 1
),
(0 1
−1 0
)]i
ch(πλ)
= ic(x+)− c(x−)
ch(πλ)
(0 1
1 0
)�= 0. (4.22)
The lemma is proved.
Proposition 4.20. The following asssertions hold.
(1) If ∞ /∈ X, then the algebra A /Ifin is isomorphic to SA .
(2) If ∞ ∈ X, then the algebra A /Ifin is isomorphic to the algebra of functions on (A /K ) \{π(∞, λ) | λ ∈ R} that are extended to functions defined on the entire spectrum of A /Kand belonging to SA .
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It suffices to take into account the isomorphism A /Ifin � Φ(A )/Φ(Ifin) and Lemma 4.18.
Lemma 4.21. Suppose that x ∈ R and Un ⊂ R is a contracting sequence of intervals with
intersection {x}, and ψn : R → [0, 1] is a sequence of continuous functions such that ψn = 1
outside Un and ψn(x) = 0. Then ψn is an approximate identity of the ideal Ix.
Proof. For each n the multiplication operator ψn is a positive element of the ideal Ix and
ψn � 1. To prove that the sequence ψn is an approximate identity of the ideal Ix, we verify the
following equality for all l ∈ Ix:
limn→∞ l(1− ψn) = 0. (4.23)
We begin with the equality (4.23) for l ∈ K . Since the right-hand side of (4.23) is linear in l
and the estimate ‖l(1− ψn)‖ � ‖l‖ holds, if suffices to verify this equality on the total set 3) in
K . For such a set we take the set of one-dimensional operators v⊗w∗ defined for v,w ∈ L2(R)
by the equality (v ⊗ w∗)f = (f,w)v. Then
(v ⊗ w∗)(1− ψn) = v ⊗ ((1 − ψn)w)∗ → 0
since (1− ψn)w → 0 in L2(R). For l ∈ K the equality (4.23) is proved.
Now, we pass to the case l ∈ Ix. As in the previous case, we can check 4.23 on a total set in
Ix. For such a set we take the set of products a1ca2, where a1 and a2 are arbitrary elements of
the algebra A and the function c : R → C is continuous and vanishes in a neighborhood of the
point x. The element a1ca2 can be represented as a1[c, a2] + a1a2c. By Theorem 3.2, the first
term is compact and, in this case, the equality (4.23) is already proved. For the second term the
equality (4.23) is valid since for some n the support of c is contained in Un and, consequently,
c(1− ψn) = 0.
Lemma 4.22. For any X ⊂ R and x ∈ R
A (X) ∩ Ix(R) = Ix(X).
Proof. We choose ψn as in Lemma 4.21 and write the ideal Ix(X) in the form
Ix(X) = {l ∈ A (X) | l(1− ψn) → 0 for n→ ∞}
and the ideal Ix(R) in the form
Ix(R) = {l ∈ A (R) | l(1− ψn) → 0 for n→ ∞}.
The assertion of the lemma follows from these equalities and the inclusion A (X) ⊂ A (R).
Lemma 4.23. A (X) ∩ Ifin(R) = Ifin(X) for any X ⊂ R.
3) A set S is said to be total in a Banach space X if linear combinations of elements in S are dense in X.
508
Proof. Using the definition of the ideal Ifin and Lemma 4.22, we find
A (X) ∩ Ifin(R) = A (X) ∩(⋂x∈R
Ix(R))
=⋂x∈R
(A (X) ∩ Ix(R)
)=⋂x∈R
Ix(X) = Ifin(X).
The lemma is proved.
Lemma 4.24. ΠXC(R) ∩ Ifin(R) = {0} for any X ⊂ R.
Proof. We consider x ∈ R and choose ψn as in Lemma 4.21. We write
ΠXC(R) ∩ Ix(R) = ΠXC(R) ∩ {l ∈ A (R) | l(1− ψn) → 0 as n→ ∞}= {l ∈ ΠXC(R) | l(1− ψn) → 0 as n→ ∞}= {l ∈ ΠXC(R) | l(x) = 0}.
Using the definition of the ideal Ifin and the above equality, we find
ΠXC(R) ∩ Ifin(R) = ΠXC(R) ∩(⋂
x∈RIx(R)
)
=⋂x∈R
(ΠXC(R) ∩ Ix(R)
)
=⋂x∈R
{l ∈ ΠXC(R) | l(x) = 0}
= {l ∈ ΠXC(R) | ∀x ∈ R l(x) = 0} = {0}.
The lemma is proved.
5 Algebra of Singular Integral Operators with Discontinuities
in Momenta and Coordinates
We fix an arbitrary pair of sets X, X ⊂ R. In this section. we study the algebra B = B(X, X)
generated in B(L2(R)) by the subalgebras ΠXC(R) and F∗(ΠXC(R)) (the homomorphism F∗is defined by the equality F∗b = FbF−1).
5.1 Ideals of the algebra B
Proposition 5.1. The algebra B(X, X) is irreducible.
509
Proof. Let Y be a reducing subspace of the algebra B(X, X). By Proposition 4.2, it has
the form χUL2(R), where χU is the characteristic function of a set U ⊂ R. We prove that U
coincides, up to a set of zero measure, with either ∅ or R. Let U be nonempty. Then there is
a nonzero nonnegative function u ∈ L2(R) supported in U . We consider an arbitrary positive
function g in the Schwartz class S(R). Then the function s =√2πF−1g is continuous on R and,
consequently, the operator F∗(s) belongs to the algebra B(X, X). But, in this case, the support
of F∗(s)u lies in U . On the other hand, this support coincides with R since
F∗(s)u =
∫
R
g(x− y)u(y)dy > 0.
The proposition is proved.
Proposition 5.2. K ⊂ B(X, X).
Proof. As is known (cf., for example, [8, Theorem 2.4.9] or [10, Corollary 4.1.10]), if the
image π(A) : A → B(H) of an irreducible representation of A contains at least one nonzero
compact operator, then K(H) ⊂ π(A). Thus, we need to verify that the algebra B contains at
least one nonzero compact operator. We consider the commutator [c, F∗(s)], where c and s are
continuous functions on R. By Theorem 3.2, it is compact. We show that it is possible to choose
c and s such that [c, F∗(s)] �= 0. For this purpose, we consider a positive function u ∈ L2(R)
and choose s equal to√2πF−1g, where a positive function g belongs to the Schwartz class, and
for c we take a nonzero nonnegative function with compact support. Then cF∗(s)u = 0 outside
the support of c and F∗(s)cu > 0 on the entire real axis.
In Subsection 4.1, we introduced the ideal Ix of the algebra A generated by functions that
are continuous in R and vanish at the point x ∈ R. In a similar way, we introduce the ideals Jx
generated by the same functions of the algebra B = B(X, X) and the ideals Jx generated by
the operators F∗(s), where s ∈ C(R), s(x) = 0. The following lemma describes the action of the
homomorphism F∗ on the algebra B(X, X). From this lemma and other assertions for the ideals
Jx we automatically obtain similar assertions for the ideals Jx. We set −X def= {−x | x ∈ X}.
The following assertion follows from the definitions of the algebra B(X, X) and mapping F∗.
Lemma 5.3. Under the isomorphism F∗ : B(X, X) → B(−X,X), the ideals Jx(X, X) and
J (X, X) are transformed to the ideals Jx(−X,X) and J−x(−X,X) respectively.
Lemma 5.4. Suppose that ψn : R → [0, 1] is a sequence of continuous functions and ψn = 1
on [−n, n], ψn(∞) = 0. Then ψn is an approximate identity of the ideal J∞.
The proof of Lemma 5.4 is obtained from the proof of Lemma 4.21 with replacing A by B,
I by J , x by ∞, and Un by R \ [−n, n].Lemma 5.5. J∞ ∩ J∞ = K .
Proof. By Propositions 4.5, 5.1 and 5.2, the ideal of compact operators is contained in each
of the ideals J and J . Let us check the inverse inclusion.
Let j be an arbitrary element of J∞ ∪ J∞. By Lemmas 5.4 and 5.3,
510
j = limm→∞F∗(ψm)j = lim
m→∞ limn→∞ψnF∗(ψm)j.
Let ψn belong to C∞0 (R). To prove the compactness of j, it suffices to establish the compactness
of the operator ψnF∗(ψm). Its kernel is equal to
g(x, y) = (2π)−1/2ψn(x)F (ψm)(x− y),
has compact support with respect to x, belongs to the Schwartz class with respect to y and,
consequently, is square integrable. These facts are sufficient for establishing the compactness of
the operator (cf., for example, [13, Theorem 11.3.5]).
5.2 Representation of the algebra B(R, R)/K as the restricted direct sum
In this subsection, we consider the case X = X = R. Throughout the subsection, we assume
that B = B(R, R) and A = A (R). We consider the following isomorphism obtained from
Proposition 2.10:
B/(J∞ ∩ J∞) � B/J∞ ⊕B/(J∞+J∞) B/J∞ (5.1)
and describe the algebras in this isomorphism. By Lemma 5.5, J∞ ∩ J∞ = K . In this
subsection, we show that the algebra B/J∞ is isomorphic to the algebra A /Ifin considered in
Subsection 4.5, whereas B/(J∞ + J∞) is isomorphic to C4.
Lemma 5.6. A + J∞ = B.
Proof. The algebra A + J∞ is a closed subalgebra of B. Therefore, it suffices to verify
that the generators of the algebra B belong to A + J∞. We have ΠRC(R) ⊂ A . Consider the
generators F∗(s) for s ∈ ΠRC(R). We write s in the form α+ β(− sgn) + s0, where α and β are
constants and s0(+∞) = s0(−∞) = 0. Now, F∗(s0) ∈ J∞ and F∗(α+β(− sgn)) = α+βS ∈ A .
The lemma is proved.
Lemma 5.7. Ifin = A ∩ J∞.
Proof. Consider an arbitrary function c ∈ Π∞C(R) with discontinuity at infinity. By
Lemma 4.19, the commutator [c, S] generates the ideal Ifin. To prove Ifin ⊂ A ∩ J∞, it
suffices to verify that [c, S] ∈ A ∩ J∞. The relation [c, S] ∈ A follows from the definition of
the algebra A . To show that [c, S] ∈ J∞, we represent the function − sgn as the sum s∞ + s0,
where s∞ ∈ Π∞C(R), s0 ∈ Π0C(R), and s0(∞) = 0. Since S = F∗(− sgn), we have
[c, S] = [c, F∗(s∞)] + [c, F∗(s0)].
The first term of this sum is compact in view of Theorem 3.2, and F∗(s0) ∈ J∞. Therefore,
[c, S] ∈ J∞ and Ifin ⊂ A ∩ J∞.To check the inverse inclusion, we show that for any x ∈ R and j ∈ J∞ ∩ A the element
j belongs to the ideal Ix. We consider a function ϕ ∈ C(R) such that c(x) = 1 and c(∞) = 0.
We have j = j(1 − ϕ) + jϕ. The first term belongs to Ix since j ∈ A , whereas 1 − ϕ ∈ Ix.
The second term belongs to J∞∩J∞ and is compact in view of Lemma 5.5. Therefore, it also
belongs to Ifin.
511
Lemma 5.8. B/J∞ � A /Ifin.
Proof. We have B/J∞ = (A + J∞)/J∞ � A /(A ∩ J∞) = A /Ifin.
The equivalence class of an element a modulo some ideal is denoted by [a]. The following
assertion is the main result of this section. Consider the diagram
B/Kp �� ��
p����
A /Ifin
q����
A /Ifin q�� ��C4 .
(5.2)
The surjective homomorphisms p, p, q, q are described by the following assertions:
(1) any element b ∈ B admits the representation b = a+ j with some a ∈ A , j ∈ J∞; for any
such a representation p(b+ K ) = a+ Ifin,
(2) if b ∈ B, then p(b+ K ) = p(F−1∗ (b) + K ),
(3) if a ∈ A , then q([a]) = (π+(−∞)(a), π+(+∞)(a), π−(−∞)(a), π−(+∞)(a)),
(4) if a ∈ A , then F∗(a) admits a representation in the form a+ j with some a ∈ A , j ∈ J∞;for any such a representation q([a]) = q([a]),
(4′) if a ∈ A , then q([a]) = (π−(−∞)(a), π+(−∞)(a), π−(+∞)(a), π+(+∞)(a)).
The values of the homomorphisms p, p, q, q on the generators are described by the following
equalities, where c and s are arbitrary elements of ΠRC(R):
p([c]) = [c], (5.3)
p([F∗(s)]) =[s(+∞)
1− S
2+ s(−∞)
1 + S
2
], (5.4)
p([c]) =
[c(+∞)
1 + S
2+ c(−∞)
1− S
2
], (5.5)
p([F∗(s)]) = [s], (5.6)
q([c]) = (c(−∞), c(+∞), c(−∞), c(+∞)), (5.7)
q([S]) = (1, 1,−1,−1), (5.8)
q([c]) = (c(−∞), c(−∞), c(+∞), c(+∞)), (5.9)
q([S]) = (−1, 1,−1, 1). (5.10)
Theorem 5.9. The following isomorphism holds:
B/K � A /Ifin ⊕C4 A /Ifin, (5.11)
defined in the sense of Propositions 2.4 and 2.8 by the diagram (5.2), which is a pullback. For
the homomorphisms p, p, q, and q in this diagram assertions (1)–(4′) and formulas (5.3)–(5.10)
hold.
512
Proof. We consider the commutative diagram
B/(J∞ ∩ J∞)r �� ��
r����
B/J∞
t����
f : [a+j] �→[a] �� A /Ifin[a]← �[a]
��
h����
B/J∞t �� ��
F−1∗��
B/(J∞ + J∞)g ��
F−1∗��
A /(I∞ + Ifin)��
ϕ
��
B/J∞
F∗
��
t �� ��
f
��
B/(J∞ + J∞)
F∗
��
g
��A /Ifin
��
h �� �� A /(I∞ + Ifin)
��
ϕ ��C4 .
(5.12)
The mappings r, r, t, t, h are the standard projections. The left upper square is a pullback
by Proposition 2.10. The isomorphisms, denoted by F∗ and F−1∗ in the diagram, are obtained
by omitting the isomorphisms F∗ and F−1∗ from the algebra B to the quotient algebras B/J∞,B/J∞, and B/(J∞ + J∞).
Let us describe the right upper square coinciding with the left lower square in the diagram.
Here, f denotes the isomorphism from Lemma 5.8. To compute its value on the equivalence
class [b] = b+ J∞, we take a representative a in the algebra A , which exists in view of Lemma
5.6. Then f([b]) = [a]. The isomorphism g is obtained by omitting the isomorphism f to the
quotient algebra
B/(J∞ + J∞) � (B/J∞)/((J∞ + J∞)/J∞
).
To verify that it is well defined, we check the equality
f((J∞ + J∞)/J∞
)= (I∞ + Ifin)/Ifin. (5.13)
The right-hand side is a subset of the left-hand side. Indeed, I∞ ⊂ J∞ and Ifin ⊂ J∞.Therefore, for any j in I∞ + Ifin
f(j + J∞) = j + Ifin.
Let us prove the inverse inclusion. We note that any equivalence class in (J∞+ J∞)/J∞ has
the form j + J∞ for some j ∈ J∞. We choose ψn as in Lemma 5.4 and write
f([j]) = f([limnψnj]) = lim
nf([ψn])f([j]) = lim
n(ψn + Ifin)f([j]).
Thus, f([j]) is the limit of elements of the ideal (I∞ + Ifin)/Ifin of the algebra A /Ifin and,
consequently, belongs to it. Therefore, the isomorphism g is well defined and the right upper
square (consequently, the left lower square) in the diagram (5.12) is commutative.
To define the mapping ϕ, we consider an isomorphism Φ between the algebra A and the
corresponding algebra of symbols SA . Using the correspondence between ideals of the algebra
SA and closed subsets of (A /K ) , we associate with the ideal Φ(I∞ + Ifin) the set
(A /K ) \ {π+(−∞), π+(+∞), π−(−∞), π−(+∞)}.
513
Thus, the spectrum of the algebra A /(I∞+Ifin) is isomorphic to SA /Φ(I∞ +Ifin), consists
of four one-dimensional representations and, consequently, the algebra itself is isomorphic to C4.
We denote by ϕ the isomorphism defined by the formula
ϕ([a]) = (π+(−∞)(a), π+(+∞)(a), π−(−∞)(a), π−(+∞)(a)) for a ∈ A . (5.14)
We denote by ϕ the isomorphism making the right lower square in the diagram (5.12) commu-
tative. It is obvious that this isomorphism is given by the formula
ϕ = ϕ ◦ g ◦ F∗ ◦ g−1.To simplify the description of this isomorphism, we note that ϕ ◦ ϕ−1 is an isomorphism of the
algebra C4. All such isomorphisms are permutations of components of C4. In order to determine
which of 24 permutations is ϕ◦ϕ−1, we compute the values of ϕ and ϕ on the equivalence classes
of elements sgn and S of the algebra A :
ϕ([sgn]) = (−1, 1,−1, 1), ϕ([S]) = (1, 1,−1,−1), (5.15)
ϕ([sgn]) = (−1,−1, 1, 1), ϕ([S]) = (−1, 1,−1, 1). (5.16)
Hence
ϕ([a]) = (π−(−∞)(a), π+(−∞)(a), π−(+∞)(a), π+(+∞)(a)) for a ∈ A .
To obtain the diagram (5.2) from the diagram (5.12), it suffices to eliminate all the algebras,
except possibly for algebras at the corners of the diagram, by replacing the homomorphisms
with the compositions: p = f ◦ r, p = f ◦ F−1∗ ◦ r, q = ϕ ◦ h, q = ϕ ◦ h. The diagram (5.2)
coincides, up to an isomorphic algebra, with the left upper square in the diagram (5.12) and,
therefore, is a pullback.
Assertions (1)–(4′) describing the homomorphisms p, p, q, q in (5.2) are obtained from the
above description of the homomorphisms in the diagram (5.12). The values on the generators
are obtained from these descriptions.
5.3 Representation of the algebra B(X, X)/K as the restricted direct sum
In Subsection 5.2, we considered the algebra B(R, R)/K . Now, we assume that X, X ⊂ R
are arbitrary. To formulate the following assertion, we need the diagram
B(X, X)/Kp1 �� ��
p2����
A1
q1����
A2 q2�� ��Cn .
(5.17)
The algebras A1, A2, number n and projections p1, p2, q1, q2 depend on the sets X and X
in the following way.
1◦. If X, X � ∞, then A1 = A (X)/Ifin(X), A2 = A (X)/Ifin(X), n = 4; the projections
p1 and p2 take the form
514
p1([F∗(s)]) =[s(+∞)
1− S
2+ s(−∞)
1 + S
2
], p1([c]) = [c], (5.18)
p2([c]) =
[c(+∞)
1 + S
2+ c(−∞)
1− S
2
], p2([F∗(s)]) = [s], (5.19)
where [c], [F∗(s)] are the generators of B(X, X)/K ; the projection q1 of A (X)/Ifin(X) is given
by the formulas
q1([c]) = (c(−∞), c(+∞), c(−∞), c(+∞)),
q1([S]) = (1, 1,−1,−1),(5.20)
where c ∈ ΠXC(R); the projection q2 onto the generators [c] of the algebra A (X)/Ifin(X) is
defined by the formulas
q2([c]) = (c(−∞), c(−∞), c(+∞), c(+∞)),
q2([S]) = (−1, 1,−1, 1),(5.21)
where c ∈ pXC(R).
2◦. If X � ∞, X �� ∞, then A1 = ΠXC(R), A2 = A (X)/Ifin(X), n = 2; the projection p1on the generators of B(X, X)/K has the form
p1([F∗(s)]) = s(∞),
p1([c]) = c,(5.22)
and the projection p2 is defined by (5.19); further,
q1(c) = (c(−∞), c(+∞)),
q2([c]) = (c(∞), c(∞)),
q2([S]) = (−1, 1).
(5.23)
3◦. If X �� ∞, X � ∞, then A1 = A (X)/Ifin(X), A2 = ΠXC(R), n = 2; the projection p2on the generators of B(X, X)/K has the form
p2([c]) = c(∞),
p2([F∗(s)]) = s,(5.24)
and the projection p1 is defined by (5.18); further,
q1([c]) = (c(∞), c(∞)),
q1([S]) = (1,−1),
q2(c) = (c(−∞), c(∞)).
(5.25)
515
4◦. If X, X �� ∞, then A1 = ΠXC(R), A2 = ΠXC(R), n = 1; the projections p1, p2, q1, and
q2 take the form
p1([c]) = c, p1([F∗(s)]) = s(∞), q1(c) = c(∞), (5.26)
p2([c]) = c(∞), p2([F∗(s)]) = s, q2(c) = c(∞). (5.27)
Theorem 5.10. The following isomorphism holds:
B(X, X)/K � A1 ⊕Cn A2, (5.28)
defined in the sense of Propositions 2.4 and 2.8 by the diagram (5.2), which is a pullback. The
homomorphisms p1, p2, q1, q2 are well defined and are given on the generators by the above
formulas.
Proof. By Theorem 5.9, the diagram
B(R, R)/Kp �� ��
p����
A (R)/Ifin(R)
q
����A (R)/Ifin(R) q
�� ��C4
(5.29)
is a pullback. Therefore, by Theorem 2.9, the following assertions hold:
(1) the homomorphisms p and p are surjective,
(2) ker p ∩ ker p = {0},(3) ker(q ◦ p) = ker p+ ker p.
We consider the diagram obtained from (5.29) by passing to subalgebras and restricting the
homomorphisms:
B(X, X)/Kp0 �� ��
p0����
p(B(X, X)/K )
q0����
p(B(X, X)/K )q0
�� �� (q ◦ p)(B(X, X)/K).
(5.30)
We show that it is also a pullback. By Theorem 2.9, it suffices to verify the following conditions:
(1′) the homomorphisms p0 and p0 are surjective,
(2′) ker p0 ∩ ker p0 = {0},(3′) ker(q0 ◦ p0) = ker p0 + ker p0.
Condition (1′) holds because of the construction of the diagram (5.30), and (2′) follows
from (2). Consider assertion (3′). The embedding ker(q0 ◦ p0) ⊃ ker p0 + ker p0 holds because
516
the diagram (5.30) is commutative. To check the inverse inclusion, we write the left-hand and
right-hand sides in the form
ker(p ◦ q) ∩ (B(X, X)/K) ⊂ ker p ∩ (B(X, X)/K
)+ ker p ∩ (B(X, X)/K
).
Using the diagram (5.12), we find the values of ker(p ◦ q), ker p, ker q:((
J∞(R, R) + J∞(R, R))/K
)∩ (B(X, X)/K
)(J∞(R, R)/K
) ∩ (B(X, X)/K)+(J∞(R, R)/K
) ∩ (B(X, X)/K). (5.31)
Below, [·] denotes the equivalence class modulo the ideal K . Consider l ∈ J∞(R, R) and
j ∈ J∞(R, R) such that [l] + [j] ∈ B(X, X)/K . We show that [l] + [j] belongs to the right-
hand side of (5.31). By Lemma 5.4, the approximate identity ψn of the ideal J∞(R, R) can be
taken in the algebra B(X, X). By Lemma 5.5, [ψnl] = 0. Therefore,
[ψn]([l] + [j]
)= [ψnj] → [j], n→ ∞,
which implies [j] ∈ B(X, X)/K . Using [l] + [j] ∈ B(X, X)/K , we find [l] ∈ B(X, X)/K .
Therefore, [l]+[j] belongs to the right-hand side of (5.31) which means that (5.30) is a pullback.
To compute the algebras p(B(X, X)/K
), p(B(X, X)/K
), and (q ◦ p)(B(X, X)/K
), we
note that the algebra B(X, X)/K is generated by [c] for c ∈ ΠXC(R) and [F∗(s)] for s ∈ΠXC(R). Therefore, their images are generated by the images of [c] and [F∗(s)].
In the case ∞ ∈ X, formulas (5.3) and (5.4) describing the values of the homomorphism
p on the generators imply that the algebra p(B(X, X)/K
)is generated in A (R)/Ifin(R) by
[c], c ∈ ΠXC(R) and [S]. Hence
p(B(X, X)/K
)=(A (X) + Ifin(R)
)/Ifin(R) � A (X)/
(A (X) ∩ Ifin(R)
)= A (X)/Ifin(X).
Here, we used Lemma 4.23.
In the case ∞ /∈ X, the algebra p(B(X, X)/K
)is generated by [c], c ∈ ΠXC(R). Therefore,
p(B(X, X)/K
)=(ΠXC(R) + Ifin(R)
)/Ifin(R)
� ΠXC(R)/(ΠXC(R) ∩ Ifin(R)
)= ΠXC(R)/{0} = ΠXC(R).
Here, we used Lemma 4.24.
Thus, for any X, X we obtain the isomorphism p(B(X, X)/K
) � A1. Using this result
and Lemma 5.3, we have
p(B(X, X)/K
)= p((F−1∗ B(X,−X)
)/K
)� A2.
Based on the known image p(B(X, X)/K
), we can compute the image (q◦p)(B(X, X)/K
)with the help of formulas (5.7) and (5.8). Replacing the algebras of the pullback (5.30) with
isomorphic algebras A1, A2, and Cn, we find the required pullback (5.17). The isomorphism
(5.28) is obtained from Proposition 2.4, and the formulas for the homomorphisms p1, p2, q1, q2follow from formulas (5.3)–(5.10) and the description of the above-obtained isomorphisms.
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5.4 Consequences
To prove the following assertion, one can apply Theorem 2.16 to the diagram (5.17) which
is a pullback in view of Theorem 5.10.
Theorem 5.11. In the notation of Theorem 5.10, the following homeomorphism holds:
(B(X, X)/K
) � A1 �(Cn) A2, (5.32)
defined in the sense of Propositions 2.5 and 2.6 by the pushout
(B(X, X)/K ) A1��
p1��
A1
��
p2
��
(Cn) .��q2��
��
q1
��
(5.33)
The set of classes of one-dimensional and two-dimensional representations of the algebra Bis denoted by (B)1 and (B)2 respectively. Then
B = {[id]} ∪ (B)2 ∪ (B)1.
Replacing in (5.28) the algebras A1 and A2 with isomorphic algebras, we obtain the isomorphism
Ψ: B/K → SB, where the algebra SB consists of functions on the spectrum of B/K .
Using the description of p1, p2, q1, q1 and the spectra of algebras A (X)/Ifin(X), ΠXC(R),
one can obtain from Theorem 5.10 a complete list of nonzero irreducible representations of the
algebra B(X, X)/K and compute the values of these representations on the generators. In the
formulas below, c is an arbitrary element of ΠXC(R) and s belongs to ΠXC(R). In the case
X, X � ∞, the list is as follows:
1. The representations p1(π±(x)), x ∈ ΠXR,
p1(π±(x)
)([c]) = c(x),
p1(π±(x)
)([F∗(s)]) = s(∓∞).
(5.34)
2. The representations p1(π(x, λ)), x ∈ X \ {∞},
p1(π(x, λ)
)([c]) =
(c(x+) 0
0 c(x−)
), (5.35)
p1(π(x, λ)
)([F∗(s)]) =
s(−∞) + s(+∞)
2Id2 +
s(−∞)− s(+∞)
2M∗(S)(λ). (5.36)
3. The representations p2(π±(x)), x ∈ ΠXR,
p2(π±(x)
)([c]) = c(±∞),
p2(π±(x)
)([F∗(s)]) = s(x).
(5.37)
518
4. The representations p2(π(x, λ)), x ∈ X \ {∞},
p2(π(x, λ)
)([c]) =
c(+∞) + c(−∞)
2Id2 +
c(+∞)− c(−∞)
2M∗(S)(λ), (5.38)
p2(π(x, λ)
)([F∗(s)]) =
(s(x+) 0
0 s(x−)
). (5.39)
Some representations are listed twice, namely:
p1(π+(−∞)) = p2(π
−(−∞)),
p1(π+(+∞)) = p2(π
+(−∞)),
p1(π−(−∞)) = p2(π
−(+∞)),
p1(π−(+∞)) = p2(π
+(+∞)).(5.40)
In the case X �� ∞, X � ∞, the list is as follows:
1. The representations p1(π±(x)), x ∈ ΠXR, satisfying (5.34).
2. The representations p1(π(x, λ)), x ∈ X \ {∞}, satisfying (5.35) and (5.36) hold.
3. The representations p2(x), x ∈ ΠXR,
p2(x)([c]) = c(∞),
p2(x)([F∗(s)]) = s(x).(5.41)
Here, the following representations are listed twice:
p1(π+(∞)) = p2(−∞), p1(π
−(∞)) = p2(+∞). (5.42)
In the case X, X �� ∞, irreducible representations of the algebra B(X, X)/K are the repre-
sentations p2(x), x ∈ ΠXR, satisfying (5.41) and the representations p1(x), x ∈ ΠXR, satisfying
p1(x)([c]) = c(x),
p1(x)([F∗(s)]) = s(∞).(5.43)
Moreover, p1(∞) = p2(∞).
The algebra B(X, X) is not commutative if and only if at least one of the algebras A1 or
A2 is not commutative. The algebra A1, as well as the algebra A2, is not commutative only
if it coincides with A (X)/Ifin(X) for X /∈ {∅, {∞}}. Therefore, the algebra B(X, X) is not
commutative if and only if one of the sets X or X contains ∞ and the other contains at least
one element different from ∞. We list the cases where the algebra B is commutative.
1. X, X �� ∞. In this case, B(X, X)/K � ΠXC(R)⊕C ΠXC(R).
2. One of the sets X or X is empty. For the sake of definiteness, let X be nonempty. We
also assume that X � ∞ (otherwise, we are in the above situation). Then B(X, X)/K �A (∅)/K ⊕C2 ΠXC(R).
519
3. X = X = {∞}. Then B(X, X)/K � A ({∞})/Ifin({∞}) ⊕C4 A ({∞})/Ifin({∞}), i.e.,it is isomorphic to the algebra C(S1) of continuous functions on the circle S1.
Such cases are referred to as “commutative.”
We denote by com(B) the closed ideal of the algebra B generated by commutators. In the
“commtative” case, it coincides with K .
Theorem 5.12. The algebra B is solvable. In the “commutative” case, the minimal solving
series has length 1 and can be taken in the form {0} ⊂ K ⊂ B. In the “noncommutative” case,
the minimal solving series has length 2 and can be taken in the form
{0} ⊂ K ⊂ com(B) ⊂ B;
moreover,
B/ com(B) � C((B)1),
com(B)/K � C0((B)2)⊗K(C2).(5.44)
Proof. A solving series cannot be shorter. because, in the “commutative” case, B possesses
representations of two different dimensions, whereas, in the “noncommutative” case, it has repre-
sentations of three different dimensions. The isomorphisms (5.44) hold since B/K � SB.
6 Abstract Index Group and the Fredholm Index of Operators
6.1 Introduction
Definition 6.1 (cf. [14, Definition 2.10]). Let A be a unital C∗-algebra. We denote by
Inv(A) the group of invertible elements of A and by Inv(A)0 the connected component of identity.
By the abstract index group of A we mean the quotient group
Λ(A)def= Inv(A)/ Inv(A)0.
In this subsection, we prove that there exists an isomorphism ΛInd: Λ(B/K ) → Zk, where
k ∈ {0, 1, 2} depends on the choice of the sets X and X. We also write out a formula for
computing the corresponding locally constant homomorphism ΛInd: Inv(SB) → Zk such that
the following diagram is commutative:
Λ(B/K )ΛInd ��
Zk
Inv(B/K )
����
Ψ �� InvSB .
ΛInd
��
(6.1)
As we show below, these formulas can be used for computing the Fredholm index of operators
of the algebra B. We recall that an operator b ∈ B(H) is Fredholm if its kernel ker b = {u ∈ H |b(u) = 0} is finite-dimensional, the image is closed, and the orthogonal complement of this image
(denoted by coker b) has finite dimension. The Fredholm index of an operator b is the number
520
ind b = dimker b − dim coker b. The product ab of Fredholm operators is a Fredholm operator
and ind(ab) = ind a+ind b. By the Atkinson theorem, the operator b ∈ B(H) is Fredholm if and
only if the operator b+K(H) is invertible in the quotient algebra B(H)/K(H). The Fredholm
index is unchanged if we add a compact term.
These facts allow us to find the Fredholm index on the set Inv(B(H)/K(H)) of all invertible
elements of the quotient algebra. The Fredholm index is constant on connected components of
Inv(B(H)/K(H)) and, consequently, is omitted to ΛInd(B(H)/K(H)). The above assertions
can be found, for example, in [8, Section 1.4]. The obtained mapping is the group isomorphism
ind: ΛInd(B(H)/K(H)) → Z (cf., for example, [14, Theorems 5.35 and 5.36]).
Remark 6.2. The mapping ind is constant on connected components of Inv(B(H)/K(H))
and, consequently, on components of Inv(B/K ). Hence it is possible to introduce a mapping
ind: Λ(B/K ) → Z, and for a ∈ Inv(B/K ) the value ind a is uniquely determined by the value
of ΛInd a. In other words, there exists a homomorphism Δ: Zk → Z such that the following
diagram is commutative:
Inv(B/K )ΛInd ��
ind����������������� Zk
Δ
��Z .
(6.2)
To compute the homomorphism Δ, it suffices to determine ind on the elements of Inv(B/K )
corresponding to the generators of Λ(B/K ) � Zk. For this purpose, we need the following
assertion [5, Theorem IV.7.3].
Lemma 6.3. Suppose that c, s ∈ C(R) and c − s, c + s do not vanish at any point of R.
Then
ind(c+ sS) =1
2πi
∫
R
d lnc− s
c+ s. (6.3)
6.2 Technical description of SB
The space (B)1 for any X, X is formed, up to a homeomorphism, by gluing finitely many
copies of ΠY R (possibly, with different Y ⊂ R)m in particular, at ∞ for ∞ /∈ Y and at ±∞ for
∞ ∈ Y .
Definition 6.4. Let i : ΠY R → (B)1 be an embedding, and let y ∈ Y \ {∞}. We set
i(y)def= {i(y−), i(y+)}
and call i(y) a discontinuity point in (B)1. We say that U ∈ SB is discontinuous at a point
i(y) if U(i(y+)) �= U(i(y−)); otherwise, we assume that U(i(y)) is defined and is equal to the
common value U(i(y+)) and U(i(y−)).
The space (B)2 consists of lines parametrized by λ ∈ R. Passing to the closure of this line,
we should add four points: elements of two discontinuity points in (B)1.
521
Definition 6.5. Let i : R → (B)2 be a parametrization of some line in (B)2. We say that
U ∈ SB has the standard form on this line if for some α, β ∈ C
U(i(λ)) = αId2 + βM∗(S)(λ). (6.4)
We denote by Z the set of functions U ∈ SB that have only finitely many discontinuities
in (B)1 and have the standard form on all lines in (B)2, except possibly for a finite number of
them.
We note that the set Z is dense in SB. This fact follows from the expressions of the algebra
SB via the algebras ΠY C(R), SA and similar assertions for these two algebras following from
Theorems 3.1 and 4.15. Any connected component of Inv(SB) is open and, consequently, there
is a function in Z in this component. We note that the function (6.4) is invertible for all λ if and
only if its limit values as λ→ ±∞ are invertible, i.e., if the values of this functions are different
from zero at corresponding points in (B)1.
The main tool for constructing the isomorphism ΛInd is Lemma 6.7 below. In the proof of
this lemma, the following known assertion is used.
Lemma 6.6. The fundamental group π1(GL(n,C)) is isomorphic to Z. This isomorphism
sends the equivalence class of path γ : [0, 1] → GL(n,C) to the number of rotations of det(γ(t))
around 0, given by the formula
1
2πi
1∫
0
ddet(γ(t)).
We choose an arbitrary function U1 in Inv(SB) ∩ Z and consider some line in (B)2. We
describe a procedure for transforming continuously the function U1 to the standard form on
this line. We introduce the local coordinates in a neighborhood of the closure of this line and
parametrize the line by ξ = th(λ). The values of U1 on this line are described by a continuous
function u1 that is defined on the segment [−1, 1] and takes the values in the set of 2×2-matrices;
moreover, its values are diagonal matrices at the endpoints of this segment. A neighborhood of
the closure of the line is taken so small that, on lines in (B)2 intersecting the neighborhood, the
function U1 has the standard form and is uniquely restored from its values on (B)1. The values
of the function on two parts of (B)1 adjoining to the line are defined by the pair (g1, h1) of
functions given on an interval (−ε, ε) that are continuous everywhere in this interval expect for
the point 0, where a discontinuity of the first kind is admissible. We choose a parametrization
such that
u1(−1) =
(g1(0+) 0
0 h1(0−)
), u1(1) =
(h1(0+) 0
0 g1(0−)
). (6.5)
Lemma 6.7. There exists a function U ∈ Inv(SB)∩Z that is described in the local coordi-
nates by the functions u, g, h and
1) U and U1 belong to the same connected component of Inv(SB),
2) U and U1 coincide outside the neighborhood under consideration,
3) g and h are continuous,
522
4) U has the standard form on all lines in (B)2 intersecting the neighborhood.
5)
ε∫
−εd ln g(x) = 2πin + ln(g1(ε)/g1(−ε)), where n is any given integer.
Proof. The proof is divided into four steps. Technical details are omitted.
Step 1. We continuously transform u1, g1, h1 to u2, g2, h2 so that g2 = g1, h2 = h1,
u2(ξ) = u1(−1) for ξ < −0.1, u2(ξ) = u1(10ξ) for ξ ∈ [−0.1, 0.1], and u2(ξ) = u1(1) for ξ > 0.1.
Now, u2 on [−1,−0.1] and [0.1, 1] is diagonal and constant.
Step 2. We continuously change the diagonal elements u2 on these segments keeping un-
changed their values at the endpoints so that the resulting function u3 coincides with the identity
matrix on less segments [−0.9,−0.2] and [0.2, 0.9]. Moreover, the diagonal elements on [−1,−0.9]
and [0.9, 1] make the same number of rotations around 0 as in the segments [−0.2,−0.1] and
[0.1, 0.2] respectively, but in the opposite direction. The fractional part of the rotation number
is uniquely determined by the values of diagonal u2 at the points −1,−0.1, 0.1, 1, whereas the
integer part can be chosen arbitrarily. At this step, the functions g and h remain unchanged.
Step 3. Stretching, we obtain the following functions, where y ∈ [0, 1]:
u4(y) = u3(0.5y), u4(−y) = u3(−0.5y),
g4(ε(−0.5 + 0.5y)) = (u3)22(1− 0.5y), g4(ε(0.5 − 0.5y)) = (u3)11(−1 + 0.5y),
g4(ε(−1 + 0.5y)) = g3(ε(−1 + y)), g4(ε(1 − 0.5y)) = g3(ε(1− y)),
h4(ε(0.5 − 0.5y)) = (u3)11(1− 0.5y), h4(ε(−0.5 + 0.5y)) = (u3)22(−1 + 0.5y),
h4(ε(1 − 0.5y)) = h3(ε(1 − y)), h4(ε(−1 + 0.5y)) = h3(ε(−1 + y)).
Choosing the number of rotations around 0 in a suitable way at Step 2, one can obtain the
desired valueε∫
−εd ln g4(x).
This result depends only on (u3)11(ξ) for ξ < 0 and (u3)22(ξ) for ξ > 0. Specifying the number of
rotations of (u3)11(ξ) for ξ > 0 and (u3)22(ξ) for ξ < 0 around 0, we finally achieve the situation
where the number of rotations of det u4(ξ) around 0 is equal to zero.
Step 4. By Lemma 6.6, the element u4 ∈ C([−1, 1],GL(2,C)) regarded as a loop in GL(2,C)
is homotopic to the trivial loop u5 defined by the equality u5(ξ) = Id2.
6.3 Case X, X � ∞
In this case, the algebra B has the form
B � A (X)/Ifin(X)⊕C4 A (X)/Ifin(X).
523
We note that the spectrum of the algebra B consists of a square some points of which are
bisected (i.e., some points are replaced with pairs of points in the sense of Definition 6.4) and
lines in (B)2 intertwining pairs of opposite bisected points (cf. Theorem 5.11 and 4.16 and also
the list of irreducible representations, p. 518). Using Lemma 6.7, one can show that, in each
connected component of Inv(SA ), there is a function that has the standard form on all lines
in (B)2 and has no discontinuities on (B)1. There is a one-to-one correspondence between the
set of such functions and the functions on the square (with the values in C \ {0}). As is known,the connected components of this set are parametrized by the number of rotations of a function
around 0 when the variable goes along the square, i.e.,
1
2πi
∫
( B)1
d ln(U(π)), (6.6)
where the integral over (B)1 is defined by the formula∫
( B)1
db(π)def=
∫
R
db(p1(π−(x)))−
∫
R
db(p1(π+(x)))+
∫
R
db(p2(π−(x)))−
∫
R
db(p2(π+(x))). (6.7)
If b has finitely many discontinuities, the integrals on the right-hand side of (6.7) are understood
as the sum of integrals over segments containing no discontinuities of b. Otherwise, the integral
on the left-hand side of (6.7) is undefinite.
To describe the homomorphism ΛInd: Inv(SB) → Z, we extend the notion of the “rotation
number” to Inv(SB). For this purpose, we define the integral over (B)2 by the following formula
(the integrals on the right-hand side are taken with respect to λ):∫
( B)2
db(π)def=∑x∈X
∫
R
db(p1(π(x, λ))) +∑x∈X
∫
R
db(p2(π(x, λ))). (6.8)
The integral on the left-hand side of (6.8) is assumed to be defined for all complex-valued
functions b that are continuous on the line in (B)2 and are constant on all such lines (except
possibly for finitely many such lines).
We define ΛInd on U ∈ Inv(SB) ∩ Z by the formula
ΛInd(U)def=
1
2πi
( ∫
( B)1
d ln(U(π)) +
∫
( B)2
d ln(det(U(π)))
). (6.9)
It is well defined since the function ln(U(π)) has finitely many discontinuities on (B)1 and the
function det(U(π)) is constant on the lines where U has the standard form and is continuous on
the remaining lines.
Lemma 6.8. The mapping ΛInd defined by formula (6.9) is continuous. Let U,U1, U2 ∈Inv(SB) ∩ Z. The following relations hold.
(1) ΛInd(1SB) = 0,
524
(2) ΛInd(U1U2) = ΛInd(U1) + ΛInd(U2),
(3) ΛInd(U−1) = −ΛInd(U),
(4) the mapping ΛInd is locally constant,
(5) ΛInd(U) ∈ Z.
The mapping ΛInd can be extended by continuity to the entire Inv(SB) in such a way that
assertions (1)–(5) remain valid.
Proof. Assertions (1)–(3) immediately follow from (6.9). Let us prove assertion (4). Let
U1 ∈ Inv(SB) ∩ Z, ε = 0.1∥∥U−11
∥∥−1. It suffices to verify that for any U2 ∈ Inv(SB) ∩ Z such
that ‖U2 − U1‖ < ε
ΛInd(U1) = ΛInd(U2).
Let U = U1U−12 . By assertion (2), it suffices to verify that ΛInd(U) = 0. We have
‖U − 1SB‖ =
∥∥U1(U1 + (U2 − U1))−1 − 1SB
∥∥=∥∥(1SB
+ (U2 − U1)U−11 )−1 − 1SB
∥∥
�∞∑i=1
∥∥(U2 − U1)U−11
∥∥i �∞∑i=1
0.1i = 1/9. (6.10)
Therefore, the values of U on (B)1 and the eigenvalues of U on (B)2 differ from 1 by a number
at most 1/9. By the definition of Z, the function U has the form different from the standard one
only on finitely many lines in (B)2. Let ψi be a function in SB such that ψi has the standard
form on all lines in (B)2, ψi = Id2 on the ith line in (B)2, where U has a “nonstandard” form,
ψi = 0 outside some neighborhood of this line, ψi(π) ∈ [0, 1] for π ∈ (B)1, and for π ∈ (B)2
ψi(π) = α(π)Id2,
where α(π) ∈ [0, 1]. We also assume that the above-described neighborhoods corresponding to
different i do not intersect. We represent U in the form
U = V0∏i
Vi, where Vi = 1SB+ ψi(U − 1SB
). (6.11)
At each point of (B/K ) , only one of Vi may differ from 1SB. Furthermore,
‖Vi − 1SB‖ � 1/9.
Therefore,
‖V0 − 1SB‖ � 1/4.
In an expression of the form (6.9) for ΛIndV0, the integral over (B)2 vanishes and, in the integral
over (B)1,
|V0(π)− 1| � 1/4.
Consequently, V0(π) cannot make any rotation around 0. Hence ΛIndV0 = 0.
525
We prove that ΛIndVi = 0. The function Vi has a form different from the standard one
only on a single line in (B)2. For the sake of definiteness, we assume that this is the line
{p1(π(x, λ)) | λ ∈ R} (the line {p2(π(x, λ)) | λ ∈ R} is treated in a similar way). Using the
principal branch of logarithm (in the integrals in (6.9) for U = Vi the expression under the
logarithm sign always has positive real part), we find
ΛInd(Vi) = ln(Vi(p1(π−(x−)))) − ln(Vi(p1(π
−(x+))))
+ ln(Vi(p1(π+(x+)))) − ln(Vi(p1(π
+(x−))))
+ ln(det(Vi(p1(π(x,+∞))))
) − ln(det(Vi(p1(π(x,−∞))))
). (6.12)
The right-hand side of (6.12) vanishes by formula (4.7) connecting the representations π(x,±∞)
and π±(x±).
Assertion (5) follows from (4), the density of ΛInd in Inv(SB), and the existence of an
element of the standard form in each connected component of Inv(SB) on all lines in (B)2(for such an element the integral over (B)2 vanishes and the integral over (B)1 is equal to an
integer). Arguing in the same way as in the proof of (4), we can extend ΛInd, which is locally
constant in a sufficiently small neighborhood of any point U ∈ Inv(SB), to Inv(SB). Assertions
(1)–(5) for the extended mapping are established by limit passage.
Lemma 6.9. The mapping ΛInd: Λ(SB) → Z : [U ] �→ ΛInd(U) is well defined and is a
group isomorphism.
Proof. By Lemma 6.8, the mapping ΛInd: Inv(SB) → Z is a group homomorphism, con-
stant on connected components of Inv(SB). Hence the homomorphism ΛInd: Λ(SB) → Z is
well defined. To show that ΛInd is an isomorphism, we show that the set of elements with
constant index is connected. In each connected component of this set, there is a function U
taking the standard form on all lines in (B)2, and the index of this function coincides with the
number of twistings described at the beginning of Subsection 6.3. In this case, the function U
can be interpreted as an invertible function on the square with a given number of twistings. The
set of such functions is connected. Therefore, the original set is also connected.
Lemma 6.10. ind(b) = ΛInd(Ψ(b)) for any b ∈ Inv(B/K
).
Proof. We choose c, s ∈ C(R) such that c(∞) = 1, s(∞) = 0, the functions c + s and
c − s are different from zero on R and make 0 and 1 rotations around 0 respectively. Then
ind(c + sS) = 1 by Lemma 6.3 and ΛInd(Ψ(c + sS)) = 1 by (6.9). It remains to take into
account Remark 6.2.
6.4 Cases X �� ∞, X � ∞ and X � ∞, X �� ∞
By Lemma 5.3, the algebra B(X, X) is isomorphic to the algebra B(−X,X). Therefore, we
can consider only the case X �� ∞, X � ∞. We recall that
B/K � A (X)/Ifin ⊕C2 ΠXC(R).
526
Lemma 6.11. In each connected component of Inv(SB), there is at least one function U
satisfying the following conditions:
(1) there are no discontinuity points of U in (B)1,
(2) U has the standard form on each line in (B)2,
(3) U is identically equal to 1 on p2(ΠXR),
(4) U(p1(π+(x))) = 1 for X �= ∅ and any x ∈ ΠXR.
Proof. We fix a connected component of Inv(SB) and consider a function U1 ∈ Z in this
component. Using Lemma 6.7, we continuously transform U1 to a function U2 that has the
standard form on all lines in (B)2. In the case X �= ∅, we can assume that
∫
R
d ln(U2(p1(π+(x)))) = 0.
Using continuous transformations, we pass to a function U3 having no discontinuity points of
the form p2(π(x)), x ∈ R. Since the function U3 has the standard form on all lines in (B)2, it
is uniquely restored from its values on (B)1. Furthermore, U3 has no discontinuities in (B)1and, consequently, it is uniquely described by the nonzero function in the space obtained from
(B)1 by “gluing” together elements of discontinuity points (cf. Definition 6.4). This space
is homeomorphic to a pair of circles joined by a segment. Therefore, U3 can be continuously
transformed to a function U4 that is equal to 1 on this segment. This means that the lemma
is proved in the case X = ∅. In the case X �= ∅, the number of rotations of U3 on one of the
circles is equal to zero, and one can require that the function U4 is identically equal to 1 on this
circle.
As in the case of the integral over (B)1 introduced in 6.3, we introduce the integral over
(A )1: ∫
( A )1,±
db(π) =
∫
R
db(p1(π+(x))). (6.13)
The integral on the left-hand side of (6.13) is assumed to be defined if the function b has finitely
many discontinuity points. We set
∫
( A )1
db(π) =
∫
( A )1,−
db(π) −∫
( A )1,+
db(π). (6.14)
The integral over (A )2 is defined by the equality
∫
( A )2
db(π)def=∑x∈X
∫
R
db(p1(π(x, λ))). (6.15)
For U ∈ Inv(SB) ∩ Z we set
527
ΛInd(U)def=
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
1
2πi
( ∫
( A )1
d ln(U(π)) +
∫
( A )2
d ln(det(U(π)))
), X �= ∅,
1
2πi
( ∫
( A )1,+
d ln(U(π)),
∫
( A )1,−
d ln(U(π))
), X = ∅.
(6.16)
We set k = 1 for X �= ∅ and k = 2 for X = ∅. As in Lemmas 6.8 and 6.9, we can prove the
following assertions.
Lemma 6.12. ΛInd can be extended to the entire Inv(SB). The extended mapping satisfies
the following relations:
(1) ΛInd(1SB) = 0,
(2) ΛInd(U1U2) = ΛInd(U1) + ΛInd(U2),
(3) ΛInd(U−1) = −ΛInd(U),
(4) ΛInd is locally constant,
(5) ΛInd(U) ∈ Zk.
Lemma 6.13. The mapping ΛInd: Λ(SB) → Zk : [U ] �→ ΛInd(U) is well defined and is a
group isomorphism.
Lemma 6.14. For any b ∈ Inv(B/K
)
ind(b) = Δ(ΛInd(Ψ(b))),
where the homomorphism Δ: Zk → Z is defined by the formula Δ(n) = n for k = 1 and the
formula Δ(n1, n2) = n2 − n1 for k = 2.
Proof. Consider an arbitrary pair of integers n1, n2 and an operator of the form c + sS,
where c, s ∈ C(R) are such that c(∞) = 1, s(∞) = 0, and
1
2πi
∫
R
(c+ s)(x)dx = n1,1
2πi
∫
R
(c− s)(x)dx = n2. (6.17)
From (6.3) it follows that the Fredholm index of the operator c + sS is equal to n2 − n1. For
X �= ∅, using (6.16), we find
ΛInd(Ψ([c+ sS])) = n2 − n1.
For X = ∅, in view of (6.16), we find
ΛInd(Ψ([c+ sS])) = (n1, n2).
To complete the proof, it remains to apply Remark 6.2.
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6.5 Case X, X �� ∞
In this case, the algebra B/K is commutative and
B/K � ΠXC(R)⊕C ΠXC(R) ={(c, s) | c ∈ ΠXC(R), s ∈ ΠXC(R), c(∞) = s(∞)
}. (6.18)
Denote by B the right-hand side of (6.18).
Lemma 6.15. The Fredholm index of all elements in Inv(B/K ) is equal to 0.
Proof. Let b ∈ Inv(B/K ). Under the isomorphism (6.18), an element b corresponds to
the element (c, s) of the algebra B. We have (c, s) = (1/c(∞))(c, 1)(1, s), which implies b =
[(1/c(∞))cF∗(s)] and ind b = − ind(c(∞)) + ind(c) + ind(F∗s) = 0.
We consider a homomorphism
ϕ : (C \ {0}) × Inv(B) → Inv(ΠXC(R))× Inv(ΠXC(R))
given by the equality
ϕ(α, (c, s)) = (c, sα/c(∞)).
Then
ϕ−1(c, s) = (s(∞), (c, sc(∞)/s(∞)).
It is clear that ϕ is an isomorphism of topological groups. Since the group C \ {0} is connected,
we have
Λ(B) � Λ(ΠXC(R))× Λ(ΠXC(R)).
It is easy to verify that
Λ(ΠXC(R)) �⎧⎨⎩Z, X = ∅
{0}, X �= ∅.(6.19)
In the case X = ∅, the isomorphism (6.19) has the form
ΛInd([c]) = ΛInd(c) =1
2πi
∫
R
d ln(c(x)).
Therefore,
Λ(Inv(B/K )) � Λ(Inv(B)) �
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
Z× Z, (X = ∅, X = ∅),
Z, (X �= ∅, X = ∅),
Z, (X = ∅, X �= ∅),
{0}, (X �= ∅, X �= ∅),
and the isomorphism ΛInd: Inv(B) → Zk is defined by the formula
ΛInd(c, s) =
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
(ΛInd(c),ΛInd(s)), (X = ∅, X = ∅),
ΛInd(s), (X �= ∅, X = ∅),
ΛInd(c), (X = ∅, X �= ∅),
{0}, (X �= ∅, X �= ∅).
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6.6 Abstract index group of the quotient algebra B/ comB
In the “commutative” case comB = K , the group Λ(B/ comB) coincides with Λ(B/K ) (cf.
Subsections 6.3–6.5). As is shown below, in the remaining cases, Inv(B/ comB) is connected.
Therefore, Λ(B/ comB) � {0}.Lemma 6.16. The mapping Inv(SB) � U �→ [U ] ∈ Inv(SB/ comB) is surjective.
Proof. Assume that
U ∈ SB, [U ] ∈ Inv(SB/ com(B)).
We set ε =∥∥[U ]−1
∥∥−1. Thenε = inf
π∈( B)1
|U(π)| .
By Corollary 4.15.1, on each line π : R → (B)2, except possibly for finitely many lines,∥∥∥U(π(λ)) − U(λ)
∥∥∥ < ε,
where
U(λ) = U(π1)(1 +M∗(S)(λ))/2 + U(π2)(1−M∗(S)(λ))/2,
π1 and π2 are points in (B)1 depending on π. We have
∥∥∥(U(λ)−1)∥∥∥ = 1/min(|U(π1)| , |U(π2)|) � 1/ε.
Therefore, the matrix
U(π(λ)) = (1 + (U (λ))−1(U(π(λ)) − U(λ)))U (λ)−1
in invertible. Let V be a function coinciding with U on (B)1 and on all lines in (B)2, except
possibly for finitely many lines, We assume that, on each remaining line, the function V is equal
to a continuous invertible matrix-valued function so that V is continuous on the closure of this
line. Since none of the assumptions of Theorem 4.15 fails, we have V ∈ SA . The function V
is invertible since all its values are invertible. By construction, U and V coincide on (B)1 and,
consequently, [U ] = [V ].
Proposition 6.17. If comB �= K , then Inv(B/ comB) is connected and
Λ(B/ comB) � {0}.
Proof. It suffices to verify that the group Inv(SB/ comSB) is connected. By Lemma 6.16,
any element of this group has the form [U ] for some U ∈ Inv(SB). By Lemma 6.9 in the
case X, X � ∞ and Lemma 6.13 in the cases X �� ∞, X � ∞ and X � ∞, X �� ∞, we can
assert that the function U can be continuously transformed to any function U1 ∈ SB such
that ΛInd(U1) = ΛInd(U). We consider a function U1 ∈ Inv(SB) that is identically equal to
1 on (B)1 and is different from the identity matrix only on one line in (B)2. The values of
U1 on this line are chosen in such a way that ΛInd(U1) = ΛInd(U). It remains to note that
[U1] = [1SB].
530
Acknowledgement
The work was financially supported by the Russian Foundation for Basic Research (grant
No. 09–01–00191).
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Submitted on October 11, 2010
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