Spectral invariance of smooth crossed products, and rapid decay locally compact groups

K-Theory 10: 283-305, 1996. 283 (~) 1996 Kluwer Academic Publishers. Printed & the Netherlands.

Spectral Invariance of Smooth Crossed Products, and Rapid Decay Locally Compact Groups*

RONGHUI JI 1 and LARRY B. SCHWEITZER 2 1Department of Mathematical Science, IUPUI, 402 N. Blackford Street, Indianapolis, IN 46202-3216, U.S.A. e-mail: [email protected] 2Department of Mathematics, Arizona State University, Tempe, AS 85287-1804, U.S.A. e-mail: [email protected]

(Received: December 1995)

Abstract. We show that all rapid-decay locally compact groups are unimodular and that the set of rapid-decay functions on a locally compact rapidly decaying group forms a dense and spectral invariant Fr6chet ,-subalgebra of the reduced group C*-algebra. In general, the set of rapid-decay functions on a locally compact strongly rapid-decay group with values in a commutative C*-algebra forms a dense and spectral invariant Fr6chet ,-subalgebra of the twisted crossed product C*-algebra. The spectral invariance property implies that the/f-theories of both algebras are naturally isomorphic under inclusion.

Key words: rapid-decay functions, locally compact groups, C*-algebras.

Introduction

The applications of C*-algebras to geometry and index theory, and the computation of t h e / f - t h e o r y of C*-algebras, have been the topics of much recent literature, e.g. [B-C][BCH] [B1] [Co] [C-M] and the references therein. As an approach to these related problems, one may adopt the view that when a C*-algebra is accompanied by a dense subalgebra of smooth functions, it is reasonable to think of the C*- algebra as corresponding to a 'noncommutative differentiable manifold' , and the dense subalgebra as 'C ~ functions on the manifold' (i.e. C ~ ( M ) if the C*- algebra is commutative and unital). For instance, the topological information of the K-theoretic index of a F-invariant geometric operator on a covering manifold IV/with the covering group F can be recaptured by the Chern character of the index, which lies in the noncommutative deRham theory, i.e., the cyclic cohomology, of a dense subalgebra A of the stabilization C~ (F)| of the reduced group C*-algebra C ' ; ( r ) .

In order that the fundamental pairing of Connes [Co] between the K-theoretic index of the geometric operator and the cyclic cohomology of the dense subalgebra A be a topological invariant of the manifold M = M / F , one must establish that the topological K- theory of r is the same as the K-theory of C~'(I') (see, e.g.

* This project is supported in part by the National Science Foundation Grant #DMS 92-04005.

284 RONGHUI JI AND LARRY B. SCHWEITZER

[C-M]). One way to do this is to show that A is spectral invariant in C~(I?) | [Bo], Lemma A.2.1 or [Ca-P], Appendix and [Scl]. (Spectral invariance means that the spectrum of every element of A remains the same in either algebra.)

The main task of this paper is to prove spectral invariance for a class of twisted smooth crossed products Se2(G, A; c~), which occur as smooth dense subalgebras of the reduced C*-crossed product C*(G, A; or). (Here G is a locally compact group acting on a C*-algebra A, g is a gauge (or length function, see Definition 1.1) on the group, and Se2(G, A; or) is the Fr6chet algebra of g-rapidly vanishing L2-functions from G to A, with e-twisted convolution multiplication, where cr is a Borel measurable two-cocycle on G (see Section 1).)

The problem of spectral invariance for smooth crossed products, and its applications, has been looked at before [BC], Appendix 1 [Bo], [Jil ], [Ji2], [Jo2], [Sc2], [Sc3], [Vii], [Vi2]. The spectral invariance of the L 1 smooth crossed product S((G, A) (which by definition consists of g-rapidly vanishing L 1 functions from G to A) is done for polynomial growth groups in [Sc3]. There has been a large amount of literature to determine for which groups G the L 1 group algebra L 1 (G) is spectral invariant in C*(G) [Pa], p. 696, p. 728. Such groups are called Hermitian (or symmetric) groups, and include all polynomial growth groups, the ax + b group, and many other groups. However, noncompact semisimple Lie groups are never Hermitian [Je]. Hence, in some important cases, the L 1 smooth crossed products Sf(G, A) are not spectral invariant.

If instead of using L l functions, we use LZ-rapidly vanishing functions, we recover many of these groups. In [Ha], [Jol], [Jo2], discrete groups such as the free groups on n generators, SL(2, Z), and many other non-Hermitian groups, were shown to have the property that the set of LZ-Schwartz functions S~(G) forms a spectral invariant dense Fr6chet subalgebra of the reduced group C*-algebra C~(G). (A group G together with a gauge g for which the inclusion S~(G) ~-+ CT(G) holds is called rapid decay (RD). See the beginning of Section 2 for examples.) In contrast to Hermitian groups, solvable rapid decay groups must have polynomial growth (see [Jol], Corollary 3.1.8 or Section 2, Example (5) below). In [Jol] and in [JoV], it was shown that semisimple Lie groups such as SL(2, ~) are rapid decay groups, as are the following classes of groups: (a) the groups of isometrics of a Riemannian symmetric space of rank one with noncompact type; (b) the discrete groups acting properly discontinuously with compact quotient on hyperbolic and geodesic metric spaces in the sense of Gromov; (c) any locally compact unimodular group acting properly on a locally finite tree, with a finite quotient graph. However, the spectral invariance of Se2(G) in the reduced C*- algebra C~(G) was not obtained in general.

In this paper, we show that any rapid-decay locally compact group must be unimodular (Theorem 2.2 below). We then show that for any rapid-decay locally compact group, the group Schwartz algebra Se(G) is spectral invariant in the reduced group C*-algebra generalizing Jolissaint's result in the discrete case. More- over, we generalize these results to the case of the twisted smooth crossed product

SPECTRAL INVARIANCE OF SMOOTH CROSSED PRODUCTS 285

Se(G, A; ~), where A is a commutative C*-algebra, a is any Borel measurable two-cocycle on G (see Theorem 2.6 below), and the group needs to be strongly rapidly decaying, which is formed by using a version of Roe algebra [Roe]. It is worthwhile to point out that the general case for crossed product algebras is much more difficult than the case for group C*-algebras. The dense subalgebra S2e(G, A; a), even as a space, depends on the action.

The task of Section 3 is then to obtain the spectral invariance of S~(G, A; a). We introduce the twisted Roe algebra B~(A) (Definition 3.2) and show that the C* completion B~(A), contains (an isomorphic copy of) the reduced C*-crossed product C~(G, A; or) as a fixed point algebra for an action/3 of G (see Proposition 3.4). Then we define an ideal J of B~(A) (Definition 3.8), and an action 7 on J involving the gauge ~ (Lemma 3.5). We show that the set of C~176 Joo, when intersected with the fixed point algebra for/3, is none other than Se(G, A; ~) when (G, g) has property RD (Proposition 3.10). Whence we obtain the spectral invariance of S~(G, A; ~) in the reduced C*-crossed product. In fact, since the spectral invariance was obtained by taking Coo-vectors, we get the stronger result that S~(G, A; or) satisfies the differential seminorm condition (see Definition 1.7, Theorem 2.5, Lemma 3.11, [B1-C]).

1. Definitions and Smooth Crossed Products

Let ( A, c~, G) be a C*-dynamical system, where G is a locally compact (Hausdorff) topological group with identity element e, and A is a C*-algebra on which G acts via &.

Recall that a Borel measurable two cocycle a on G is a Borel measurable map or: G • G --+ T such that ~(a, b)a(ab, c) = or(a, bc)a(b, c), a(e, b) = a(a, e) = I, and ~(a, a - 1) = l , for all a, b, c E G. It is well-known that this last condition can be made within the cohomology class of a when the measurability of ~ is not required. From these conditions one obtains the extra condition: a(a , b) = :r(b -1, a - l ) . In fact, cr(a,b) = cr(ab, b -1) = cr(b-la-l,a) : a (b - l , a - l ) .

Let L~(G, A) denote the set of compactly supported Loo functions from G to A (with compact supports up to null sets). (See [Tr], Sect. 46.1, p. 468, for the definition of L~176 A), where K is a compact subset of G. Then define L~(G, A) to be the union (or inductive limit) of the Loo(K, A), where I f ranges over all compact subsets of G.) Then L ~ (G, A) is a ,-algebra under the a-twisted convolution product

and involution

286 RONGHUI Jl AND LARRY B. SCHWEITZER

Assume that A is ,-represented faithfully on a Hilbert space 7-/. Then L~(G, A) is ,-represented as a ,-algebra of bounded operators on the Hilbert space L2(G, 7{) via the formula

~(t) = fG a t - ' (~2(s) )~(s- l t )a( t - | ' s) ds, *a

where ~ E L~(G, A), and ~ E LZ(G, 7-[). The reduced and a-twisted crossed product C*-algebra C*(G, A, ~; a) is defined to be the completion of L~(G, A) in this representation. We will often suppress the a from the notation.

Note that we take the completion of L~(G, A) rather than of the compactly supported continuous functions Cc(G, A), because the latter may not be closed under the a-twisted convolution if a is not continuous. However, since C~(G, A) is dense in L~(G, A) in the C* topology, the completions will be the same if a is continuous.

For further details of the reduced crossed product C*-algebra of A by G, we refer the reader to [Pd], and to [Z-M] for the a-twisted reduced crossed product C*-algebra of A by G, in the case that G is countable and discrete.

Remark. Throughout this paper, A will be the modular function on G which satisfies the conventions

f(hg) dh = A(g) -1 f f(h) dh,

f f(h-1) dh = f A(h)-l f(h) dh'

for f E co(G).

DEFINITION 1.1. A Borel measurable function g : G --~ [0, ~ ) which satisfies 6(gh) < 6(g) + 6(h), 6(g - | ) = 6(g), and 6(e) = 0 is called a gauge on G. (Gauges are also called length functions in [Jol] [Jil]. See [Pr], Sect. 2 and Example 4.2(2) for more on the terminology.) Gauges are automatically bounded on compact sets [Sc4], Theorem 1.2.11. A gauge 61 strongly dominates another gauge 62(62 <_s 61 ) if there exists d E N and c > 0 such that

62 (g )_<c( l+g1(g ) ) d, g E G.

The two gauges 61 and 62 are strongly equivalent if61 <s 62 and Q _<~ 61. If G is compactly generated, with open and relatively compact generating set

U satisfying e = l a E U, U = U -1, and U~=o Un = G, then the word gauge ~u(g) = min{nlg E U ~} is a gauge on G. The strong equivalence class of 6u is independent of the choice of generating set U, and the word gauge strongly dominates every other gauge on G [Sc4], Theorem 1.1.21. A special case of this definition is the word length function on a finitely generated discrete group, where U is a set of generators together with their inverses. We will be considering general gauges 6 throughout this paper, and will say explicitly if 6 is to be the word gauge.


DEFINITION 1.2.By a Fr~chet algebra we mean a Fr6chet space (over the com- plex numbers) with an algebra structure such that multiplication is jointly continuous [Wa], Chapter VII. By a Frdchet *-algebra we mean a Fr6chet algebra with a continuous involution defined on it. A Fr6chet algebra ..4 is said to be m- convex if there exists a generating, increasing set of seminorms { 11 �9 [1~ } which are submultiplicative. That is, for each n,

Ilablln Ilall " Ilbll , a,b A.

Let A be a Fr6chet algebra with a continuous action a of G.

1.3. Let G and g be as in (1.1) and A be a Fr6chet ,-algebra upon which G acts as a group of .-automorphisms via a : G ~ Aut(a) , let {[]. [[~} be the increasing sequence of seminorms on A. Set Sez(G, A) to be the set of all L z- Schwartz functions from G into A with respect to the gauge g in the sense that S~(G, F ) is the set of functions ~: G ~ A satisfying

IIl lll ,k= + (g))2k dg) n < 00,

for all n, k = 0, 1,2, .... Then S~(G, A) becomes a Frdchet space with the topology generated by the seminorms {11]. [[]~,k}. It is clear that A can be any Banach or C*-algebra. In the case when A is a Banach algebra, we will denote the seminorms simply by []]~[][k, f o r ~ E S~(G,B). When A is C, we will write S~(G) instead of c).

If A is a Fr6chet ,-algebra with e-tempered action of a unimodular group G by ,-automorphisms, then the involution ~*(9) = ag(~(9-1)*), defined at the beginning of Section 1, gives a conjugate linear automorphism of the Fr6chet space Se2(G, A).

Remark 1.4. In this paper we only deal with the case when A is a commutative C*-algebra. The general case will be studied in the future.

DEFINITION 1.5. Let ~ : A ~ A be an algebra homomorphism of an algebra ,,4 to a Banach algebra A, with dense image. (Assume that L(I4) = 1A if A is unital.) We say that A is spectral invariant (SI) in A if every a E A is invertible in A if and only if L(a) is invertible in A [Scl] [Bo], Appendix. It follows that ,,4 is SI in A if and only if Mu(A) is SI in M~(A).

If A is a Frdchet algebra, we require the map ~ : A ~ A to be continuous. (In this case, if ~ is injective (as it usually will be in this paper), spectral invariance is equivalent to Mn(A) being closed under the holomorphic functional calculus in M,,(A) for each n [Scl], Corollary 2.3, Theorem 2.1. In the terminology of [Jil], this means that A is smooth in A.)

Let ~ : A -~ A be a continuous algebra homomorphism of a Frdchet algebra A to a Banach algebra A, with dense image. Then A satisfies the differential


seminorm condition (DSC) in A [B1-C], [Sc3] if there exists a family of seminorms {ll �9 [l~}~=0 for A and a constant c > 0 such that

]lab[Ira c Ilallillblls, a,b A, i+j=rn

where I1" II0 : II~(')IIA is given by the Banach algebra norm on A. (The notation Ei+j=.~ is short for ~i~0, with j = m - i, ) We say that A is strongly spectral invariant (SSI) in A [Sc3], Section 1, [Sc2] if for every m, there is some p,~ >_ m, c > 0, and D.~ > 0 such that

II a, . . . an l l~ _< cnDm ~-~ Ilalllk,. �9 Ila,~[Ikn,

for all n-tuples al, . . . , an E r and all n, where the sum is over those k~s such that ~r 1 ki < Pro. With this definition we have the implications DSC =~ SSI ~ SI. ([Sc3], Proposition 1.7, Theorem 1.17.)

2. Property RD, SRD, Examples, and the Main Theorem

DEFINITION 2.1. Let G be a locally compact group with gauge g. We say that (G, e) has property RD (or is rapid decay) if the LZ-Schwartz functions S~(G) are contained in C~(G) [Jol], 1.2. If we simply say that G has property RD, this means that there exists a gauge g on G such that (G, g) has property RD. Note that if (G, gl ) has property RD, and C1 is strongly dominated by another gauge g2, then (G, 62) has property RD. Hence if G has property RD and is compactly generated, then (G, gu) has property RD, where gu is the word gauge on G (see Definition 1.1).

EXAMPLE 1. Let G be a discrete group. According to [Jol], G has property RD if G has polynomial growth [Grl ] or if G is a hyperbolic group. Moreover, the word hyperbolic groups of Gromov [Gr2] are RD-groups [Ha]. In [Jol], Jolissaint also showed that the amalgamated free product of two RD-groups by a finite subgroup is once again an RD-group; and in [JoV], Jolissaint and Valette showed that a discrete group acting properly discontinuously with compact quotient on a hyperbolic and geodesic metric space in the sense of Gromov is rapidly decaying (this class is bigger than the word hyperbolic groups). This provides a good class of discrete RD-groups. See also [Jol], Example 2.1.11 (braid group), and for non-RD discrete groups see [Jol], Corollary 3.1.9 (SL(n, Z), n _> 3), Example 1.2.8.

EXAMPLE 2. Let G = F ~ be the free group on countably infinitely many generators. Then G is not compactly generated, so there is no word gauge on G. (Moreover, there is no gauge that dominates every other gauge on G [Sc4], Example 1.1.15, Theorem 1.1.21.) It is well known that there is an injective group homomorphism of G into F2, the free group on two generators. Let g be the word


gauge on F2. Then, as we noted in Example (1), (F2, g) has property RD. Since G is a subgroup, it follows that (G, (la) also has property RD (by an easy argument, or [Jol], Proposition 2.1.1). Note that this is one reason we must include gauges other than the word gauge in the definition of property RD.

EXAMPLE 3. Since SL(2, R) has real rank one, it has property RD by [Jol], Corollary 1.4 of the Appendix. Examples of this kind can also be found in [JoV], in which the authors proved more generally that the group of isometries of a Riemannian symmetric space of rank 1 of noncompact type is rapidly decaying.

EXAMPLE 4. In [Vi2] and [Vil], Proposition 28, Theorem 29, there are many examples of Lie groups which possess properties similar to property RD.

EXAMPLE 5. Any locally compact group of polynomial growth is an RD-group with respect to the word gauge. (Polynomial growth means that G is compactly generated with generating set U, and that the Haar measure I Unl is bounded by a polynomial in n.) Moreover, an amenable locally compact group has property RD if and only if it has polynomial growth. (Proof The same arguments as Theorem 3.1.7 and Corollary 3.1.8 of [Jol] can be carried over in the locally compact case without change.) Note that all solvable locally compact groups are amenable [Pa], p. 729.

The following is an explicit example of a non-RD, connected Lie group.

EXAMPLE 6. Let G be the ax + b group

{ ( ; a ~ ) a , bEIR}.

We show that G does not have property RD. (In contrast, L 1 (G) is spectral invariant in C,.*(G)[Pa], p. 696 or Example 21.) Note that the Haar measure on G is given by e -a d a d b, where da db is Lebesgue measure on R. The word gauge on G is strongly equivalent to the gauge

r(g) = ln(1 + e la[ + e-albl + Ibl),

[Sc4], Example 1.6.1. Let p: R 2 + R + be defined by

ea/2e-I~l/10 p(., b) _

(1 + Ibl)3/4

Then p 6 S](G) , since

ea/2e-lal/10x~ 2

e-ada db


i f e-tal/]3/2Polyk(lal,ln(1 -< (1 + Ibl + Ibl ) )dadb <- Ck < oo.

We show that p , p g~ L2(G) by showing that p �9 p(a, b) does not converge for any (a, b) C IR 2 for which a < 0. (Here �9 is meant to denote the left regular representation on L2(G).) In fact,

p* p(a,b) = f i P (u ' v )p (a - u, e-U(b _ v))e-" dudv

i S eU/2e-lul/l~ e(=-~)/2e-b-~l/ l~ = (1 + iv1)3/4 ( 1 +]e -u (b - v ) ] )3 /4e -~dudv

eal2rbr"]ooT- e~/l~ e(U-a)/l~ > e -~ du dv - ~ (1 + Ivl)3/4 e -3U/4-~ -~7 v1)3/4

i b vl)gl4dV i f ~ = e2a/5 1 �9 e -u/2~ du.

~-~o (1 + Iv1)3/4(1 + I b -

Since f a e_U/20 du does not converge for any a < 0, p * p is not defined, and so

p . p r L2(G). Hence, p ~ C:(G), and G does not have property RD.

This example is a special case of the following theorem (and also of the previous example).

THEOREM 2.2.Assume that G is a locally compact group, which is not unimodular, and let g be any gauge on G. Then (G, g) does not have property RD. Moreover, (G, g) cannot be made to have property RD by changing the convention on the involution to be ~*(g) = ~(g-I)A(g)-~ for some a E R other than a = 1.

Proof. The proof is similar to the argument in [HR], (20.34) p. 308. For the reader's convenience, we will give a complete account of this.

Since G is non-unimodular, there exists an a E G such that A(a - l ) _> 2. Let U, V be symmetric neighborhoods of the identity element e C G, so that U is relatively compact, U 2 C {x E G : 1/2 < A(z) < 2} and V 2 C U. Note that since gauges are automatically bounded on compact subsets of G [Sc4], Theorem 1.2.11, we may assume c > 0 is such a bound for g on U.

Now the sequence of subsets { ak U } ~=_~ are pairwise disjoint. For k = 1 ,2 , . . , choose symmetric neighborhoods Wk of e, such that Wk C V and the Haar measure A(Wk) = 2 -k. (This can be done since G is nonatomic.)

Let nk 3k + 2. Set W = ,~oo .~k w . = ,~oo a~k A(W) = ~k-I ~ , , x and A Ua k. Then = - "-'k= 1

E~--I A(Wk) = E~=l 2-k = 1. Define ~ = X w, and f = A - I / 2 ) s We show that both c 2 and f are in S2~(G),

but f �9 c 2 ~ L2(G). Thus, G cannot be rapidly decaying. In fact, for any positive integer m,

[ io2(g)12(1 + ~.(g))2mdg = [ x w ( g ) ( 1 + e(g))2mdg J J


_<

Er176 fa'~kWk (1 + e(g))2m dg k=l

Z~176 ~kWk (1 § nkg(a) + c) 2m dg k=l

o o

E k = l

2-~(1 + nke(a) + e) 2m < ec,

since nk = 3k + 2. Similarly,

if(g)12(1 + •(g))2m dg = f A(g)-lxA-,(g)(1 + ~(g))2"~ dg

= fA(i + e(g))2mdg

o o

<- E A(ankUak)( 1 + c + (nk + k)~(a)) 2m k=l

o o

= E A(U)A(ak)( 1 + c + (4k + 2)e(a)) 2m k=l

o o

-<E k=l

A(U)2-k(1 + c + (4k + 2)e(a)) 2m < ~ .

Now we prove that f �9 ~2 ~ L2(G) �9 Let

~(~) = f * p(x) = f f(g)~(g-~x) dg

= / A(g)-I/2XA-~ (g)xw(g-lx) dg

= f A(g)-U2XA(g-l)xw(g-lx) dg

= f A(g)-~/2~A(g)~w(gx) dg

= f A(UX-1)-I/2~A(UX-1)~W(U)A(X)-I du

= / A(X)-I/2/k(U)-I/2~A(UX-1)XW(U ) du.


For u E a'~k Wk and x C a-kV, we have ux -1 E A. Therefore,

l(x) _> ~r~kWk A(x)-l/2A(u)-l/2 du _> 2kA(Wk) = 1.

The last inequality can be estimated as follows. Let x = a-kv, u = a nk wk, where v E V, wk E Wk. Then

A(X)-I/ZA(u)-I/2

= A(a-kv)-l/2n(ankwk) -1/2 :> 1A(a)(-nk+k)/2 > 2 k. (2.1)

Since A(U~~ = oo, f �9 ~2 ~ L2(G) �9 This completes the proof of the first statement. The second can be proved by

modifying the proof above. []

DEFINITION 2.3.Let (X, d) be a locally compact, metric space with a Borel measure > on X. Let LZ(x ) be the natural LZ-space of X with respect to the measure.

(a) Let B S z ( X ) be the Frech6t space of functions k on X • X satisfying

Ik(x'y)12(1 + d(x,y))2kdy supxex ex

Ik(x,y)12(1 +d(x,y))2hdx < oo, k = 1,2, + SUpYex eX

The seminorms I1" IIs are defined by

Ilklls = (sup Ik(x,y)[2( 1 +d(x'y))ekdy ) 1/2 \xeX eX

+ ( s u p f~eX Ik(x' y)I2(1 + d(x, y))2k dx) 1/2, yEX

where k E BS2(X) and s = 0, 1,2, . . . . (b) Given a function k : X • X --+ C. k is said finitely propagated if there is a

constant ck > 0, such that, k(x, y) = 0 whenever d(x, y) > ek. k is said bounded if it defines a bounded operator on L2(X) by convolution. That is, k : L2(X) --+ L2(X) defined by

k * ~(,) = f k(x, y)~(y) dy Jy ex

is a bounded operator. The precompleted uniform Roe algebra of X is defined to be

B~ = {k" X • X ---+ C [ k is bounded and finitely propagated}.

L e t / 3 ] be the norm closure of B ] in L2(X), and is called the uniform Roe algebra.

(c) X is said rapidly decaying i fBS2(X) C B] .


Remark 2.4.It is clear that B~, C BS2(X). Also by the closed graph theorem (X, d) is rapdily decaying if and only if there exist a C > 0 and a s > 0, such that I{~[l~j{. -< C]]P]I~ for all p r BS2(X).

DEFINITION 2.5.A topological group G with a length function ~ is said to have property SRD (strongly rapidly decaying) if BS2(G, C) is contained in Bo(C).

Since S~(G) is contained in B S 2 ( G ) as those kernal operators supported only on the diagonal and the same is true for C,*.(G) i n / ) ~ , property SRD implies property RD. The following theorem is due to G. Yu and the first author.

THEOREM ([Ji-Yu]). Word hyperbolic groups and groups of polynomial growth ha ve proper O, SRD.

The following theorem and corollary are the main results of the paper.

THEOREM 2.6. (a) If(G, ~) has property SRD, and (i' acts on a commutative C*- :r f'~ f algebra A, then Sg2( G, A; a) is a Frdchet -subalgebra of C,, ( G, A; or) and satisfies

,g the differential seminorm condition in C~(G, A; or). In particular, 52(G , A; a) is a spectral in variant dense Frdchet subalgebra of C; ( G, A; cr ).

(b) When A = C and (G, f) satisfies the property RD, 5'~(G; or) is a sepctral invariant Frdchet ,-subalgebra of C*(G).

Remark 2.7. (a) If the theorem holds, then C~(G, A; or) is necessarily the envelop- ing C~-algebra of S~(G, A; or) [B1], 3.1.3, 3.1.5.

(b) The Frdchet algebra S~(G, A; or) in Theorem 2.6 is necessarily 'm-convex, since the DSC implies m-convexity [Sc3], Proposition 1.7.

We give some examples to illustrate the main theorem.

EXAMPLE 7. Let G be any word hyperbolic discrete group with word gauge (~ [Gr2]. Let M be a locally compact space on which G acts. (For example, we could let M = OG, the boundary, and by [Gr2] M is compact.) Then G acts on ( '0(M) (continuous functions on M vanishing at infinity) by translation, and ,5'~( G, (-. 0(~ J )) satisfies the DSC in C r (G, CO( ; I )) by Theorem 2.5. Alternatively, let G' = SL(2, R) act on the boundary of the hyperbolic plane M = OH> Then we get the same result.

We give an example which illustrates how the e-temperedness condition can be restrictive.

EXAMPLE 8. Let G = SL(2, R) act on the hyperbolic plane H2, and let ,~ be the word gauge on G. By Theorem 2.5, we know that S~(G, Co(H2)) is spectral invariant in C~*(G', Co(He)), by the DSC. We would like to replace C'o(H2) with


some smooth subalgebra S(H2) of functions on H2, and apply Corollary 2.7 to see that Sze(G, S(H2)) is spectral invariant in Cg(G, Co(H2)). Note, however, that Corollary 2.7 requires that the action of G on S(H2) be g-tempered. If we take S(H2) to be rapidly vanishing continuous functions on H2 (where rapidly vanishing means with respect to the scale on H2 inherited from the gauge t~ on G), then the g-temperedness condition would be met, and we can apply Corollary 2.7. On the other hand, if one tries to let 5'(H2) be G-differentiable functions, one runs into problems. We usually obtain the g-temperedness on sets of C~176 from [Sc4], Theorem 4.6, which assumes that g bounds Ad. But for C to bound Ad, it is necessary that G be a Type R Lie group [Sc4], Theorem 1.4.3. This is not the case for G = SL(2,R). So one must be satisfied with using for S(H2) algebras of continuous or rapidly vanishing functions, at least without further analysis.

3. Proof of Theorem 2.6

We prove Theorem 2.6 in a series of definitions and lemmas.

DEFINITION 3.1. If A is any C*-algebra, then LZ(G, A) is a Hilbert module over the ,-algebra L~(G, A) with the action defined by

~(g) = / c~g-, (~(h))~(h-lg)a(g-l, h) dh. *a

The operator norm on L~(G, A) is then a C*-norm, and we let C~(G, A; ~) denote the completion. Note that if A = C, then C~(G; a) = C*(G; a). In general, by Rieffel's lemma [Ri], Lemma 2.4, C;(G, A; a) ~ C:(G, A; a). In fact, equip LZ(G, A) with the A-valued inner product:

(~, ~])A = J~ ~(g)*q(g) dg.

By a straightforward computation, ( ~ , ~ , Z/)A = (~, ~**~ 7/)A for ~ E L~(G, A), so all elements of L~(G, A) have bounded adjoints. Hence, L2(G, A) is a nilbert C; ( G, A; a)-module. By [Ri], Lemma 2.4, we have C; ( G, A; a) ~- C*( G, A; a ).

CONVENTION. In what follows, for an element a E A, we will use the notation [[a]] = a*a for notational convenience.

DEFINITION 3.2.Let A be a C*-algebra upon which the group G acts by a. Let B c ( A ) = those kernel functions k : G • G ~ A with k E L~(G • G, A) (the set of all Loo functions on G • G with values in A), such that the support of k is contained in a bounded neighborhood of the diagonal of G • G. The last condition amounts to saying that k(x, y) = 0 if xy -1 ~ K, for some compact subset K of G, which depends on k.

We make Bc(A) into a ,-algebra with multiplication

k ,~ l(x,y) = [ k(x,z)axz-,(l(z,y))a(xz-l,zy-1)A(z) -1 dz, JG


and ,-operation

(/~:)*(x, y) = OQL.y-I (~,~(y.~ X)*)/~k(yx-l).

Note first that the cocycle conditions imply the following identities: a ( x - 1 x y - l ) =

or(x, y-l), and a(x, y-l)o-(y, X - l ) : 1. Note also that in the integral expression for multiplication, the integrand is zero unless xz -l and zy -l both lie in some compact set K depending on k and 1. Hence, for each x and y, z is restricted to a compact subset of G and the integral makes sense. Also, the integral is nonzero o n l y i f x y -I E K2, sok*c~l E BG(A).

We define the twisted kernel operator h a on LZ(G, A) associated with the kernel I,: ~ Bc;( A) by

= s *v-l)A(v) -1 dy.

One checks that under the map k --+ k s, the image of BG(A) is a ,-subalgebra B~)(A) of bounded Hilbert A-module maps with involution�9 In fact k + k ~ is a ,-algebra homomorphism, with respect to the A-valued inner product (., ")m on LZ(G, A) defined in (3.1). Also, k ~ is a bounded operator by the following estimate, which uses the definition of k ~ % ~ above.

�9 2

[ , k ~ , ~ , ] 2 _< sup j(j]lk(x,y)HA[[A-l~(y),iHsdy) dx fGS(A)

o /(/ ) _< M sup ~1r dy dx f E S ( A )

_~r sup / ( / -1 )2 = xlr x)liHy dy dx fGS(A) .

<__ M#(K):II tI , where IiaiiHs is the Hilbert space norm of a E A in the Hilbert space Hf associated with the GNS construction for a state f on A, and where Xh denotes the characteristic function of the set K, which is compact and so has finite measure.

We would like to call this algebra, B~(A), an (c~, a)-twisted Roe algebra with coefficients in A. The standard Roe algebras were constructed (using continuous functions instead of L~176 and without twisting nor coefficients) and studied by J. Roe in his study of index theory of geometric operators on an open manifold. Elements in Roe algebras correspond to kernel operators of finite prop- agation speed [Roe].

The closure of B~(A) is denoted by B~(A). There is also a natural right G- action on B~(A) given by the diagonal action t3~(k)(x, y) = k(xg, yg) on B~(A). Let B~(A) ~ be the G-invariant part of B~(A). An element k E B~(A) ~ satisfies


k(xg, Yg) = k(z_, y) for all 9 E G. This, in particular, implies that k(xy -~ , e) = k(x, y). We set k(x) = k(x, e) for k E B~(A) ~.

LEMMA 3.3./3 gives an action of the group G on the space B~(A), which is isometric for the operator norm on B~( A ). The map g ~ /3g is a homomorphism from the group G into the group of .-automorphisms of B~( A ).

Proof In order to show that/3 is isometric, it suffices to show that k and/3~(k) have the same norms as operators on LZ(G, A). For ~, ~/E LZ(G, A), set

~x(z) = ~ - , ( ~ ( z x - ' ) ) ~ ( x , z - ' ) A ( x ) -~/2

and

~x(y) = ,x- , (~(yx-1))~(x , y-~)A(x) - ' / : .

It is easy to check that (~z,~x) = ax- , ( (~ ,~)) , and similarly for ~l. Hence, ~ ~/~ have the same norms as ~, ~l, respectively, in L2(G, A). Noticing that ~(~, y-~)~(~y-~, y~-~) = ~(~, z -~)~(y-~ , s~) , we have

dy

,]( y x - l )d( zx-1)d( yx -1)

and so /3~(k) has the same norm as k as an operator on L2(G, A). To prove the second statement, we need only to show that, for each g E G,/3g is a , - automorphism. In fact,

= / a k(xg, z)a~:g~-,(l(z, yg))a(xgz- ' , zg- ly -1)A(z ) -I dz

yg))a(xu-| , uy- l )A(ug)- ld(ug)

= f k(xg, ~9)-~-,(t(~g, yg))~(x~ -~, ~y-~)~,(~)-' d~ JG

=/3~(k) ~ ,~/3~(~)"(x, y),

SPECTRAL INVARIANCE OF SMOOTH CROSSED PRODUCTS

and

v) = y.q) = (k(yg, xg)*) X(yx = y ) . []

297

PROPOSITION 3.4. Let ,8 be as in Lemma 3.3. Then the fixed point algebra C~ 3 BG(A ), is naturally isomorphic to C; ( G , A; a) = C; ( G, A; c~ ).

t xa (2)0 Proof. It is clear that B~(A) ~ = L~ (G, A) in a natural way. In fact, if ~ C Liy'(d;, A), then k~(x, y) = ~ ( x y - l ) is obviously ~-invariant and contained in L:~'(G • G,A). I f ~ has compact s u p p o r t K , k~(x,y) = 0 i f x y -I ~_ K, so

to.;. E Ba(A). On the other hand, given k E B~(A) 9, let/c r L ~ ( G , A) be defined

as L'(y) = k(y, c). Since k(y, e) = 0 if y ~ Kk, k has compact support, and so lies in L~(G, A). The correspondences k ~ ~" and ~ ~ k~ thus give a bijection 13~i(A)/3 ~- L~(G, A), which is easily seen to be a , -algebra isomorphism.

By invariance, k(x, y) = k(xy -1, e), for all x, y E G. Hence, for~ E L2(G, A), le ff B ~ o ( A ) ~, we have

k ~ ,~ ~(x) = ~ ~z-j(k(x, y))A-1~(y)cr(x - ' , x y - l )dy

= / a ct~.-, (k(xy -1, e ) )A- l~ (y )c r (x - l , xy -1 ) dy

= ~ozx - , ( k ( y ) )~ (y - l x )a ( x - l , y )dy = k *~ ~(x),

so IFklIB~(A)~ -- IiklIc;(G,A, /, since the topologies on both C*-algebras are given

by the operator norm on L2(G, A). This implies that B~(A)# ~ C~(G, A; a) . []

L E M M A 3.5.For k C B3( A ), define 7t( k )( x, y) = exp( it( g( x ) - g(y) ) ) , k( x, y) ]br t E R. Then 7t extends to a strongly continuous action of IR on the C*-algebra B3(A).

Proof. For k e B3(A ), ~,,1 e LZ(G,A), we have (T t (k )*~ ~,,1)A = {# *~ ~, f?}A, where ~(y) = exp(-ite(y))~(y) and f/(x) = exp(-i te(x))v(x). Also, ~, f /have the same norms as ~, r] respectively in L2(G, A), and so 7 is norm preserving on B~(A). It is also straightforward to check that for k, l r B~(A), %+t(k') = %(Tt (k) ) , 7t(k*) = 7t(k)*, and 7t(k *~ l) = ~/t(k) *~ 7t(1).

It remains to show the strong continuity. For k E B~(A), k(x, y) = 0 if x y - ' ~ I f for some compact subset K of G. Therefore, for such x, y in the support, [e(x) - e(y)l _< e(xy -1) _< C, since • is bounded on compact sets. Therefore, for t, s E IR such that I s - t[ _< e, we have

]exp(it(g(x) - g(y))) - exp( is (g(x) - g(Y)))I -< (~C)/(27c).

298 R O N G H U I JI A N D LARRY B. SCHWEITZER

Hence, for ~ C L2(G, A),

: sup /c ffc (eit(e(~)-e(y))-ei~(e(~)-e(y)))a~-'(k(x'y)) fGS(A)

X ~ - l e ( y ) ( Y ( x - l v x y - I ) dy 2 dx 14 ]

2

- 2rr fGS(A) JG y-leK

< (2C)(lllklllB-7~) 2 sup Illqill 2, feS(A)

where Ikl(~, y) = Ilk(*, Y)IIA, IqAx) ---- II~(*)ll,v~" Clearly, Ikl ~ BG(C), and ]~lf E L2(G). Moreover, sup/6s(A)Hl~lfl] 2 = I]~c]] 2.

Nowweshowtha t i f [ ]k~-h~[ ] < C, f o r s o m e C > 0,then []Tt(k~-h~)]] < C. In fact,

I I > ( k ~ - t~o) *~ ~ll~

= sup f c fGeit(eC~)-e(V))c%-'(k(x'Y)- h(x,y)) fCS(A)

• ~ x ( Y ) - ~ ( Y > ( x - ~ , x Y - ' ) d Y 2 dx H I

= ii(k ~ _ h ~) , . 6112,

where {t(Y) = eitg(Y)~(Y) �9 Since II~tll2 = (i~tll2, the lemma is proved. []

We would like the C~176 for the action of 7 on B~(A) to be equal to 5'2e(G, A; or), when intersected with C~(G, A; a). However, we can only prove this if G is discrete (see Proposition 3.10 below). (For a counterexample, consider the circle group (Remark 1.11).) We will therefore replace B~ (A) with an appropriate dense ideal in order to make things work out. We begin with a definition.

L 2'0o(~ A) the Banach space L0O(G, 2 , L,(G,A), DEFINITION 3.6.We denoteby /,,~v,~, 2 A- t /2 ) , the set of L~176 from G to L~(G, A), scaled against A -1/2. We

L 2,0ott'2 A) by define the norm I1" 112,oo on ~2 C A,~v,~,

[]~I[2,0o = esssup{IIF(y)U2,~A(y)-l/21y E G}.

SPECTRAL INVARIANCE OF SMOOTH CROSSED PRODUCTS 2 9 9

Here, L~(G, A) is the Banach space with the norm defined by

II'lll ~ = II f~ ~=-'(~(x)*~(x))dxllA,

r 2'~ t ~ A) as a left Banach module. THEOREM 3.7. B~( A ) acts on ~ /x,~ ,~ , L2,~176 A ~ Proof. Let ~ E A,,~,-,, j. We assume that ~ is given by a function ~(x, y)

on G x G, such that for each y E G, ~(., y) is in A). Let k C L~(G, B3(A ). We define

ko *. ~(x,y) = jfGk(X,Z)O%z-l(~(z,y))o'(xz-l, zy-1)A(z) -1 dz.

It is easy to verify the cocycle identity:

(7(XZ-' 2:y -1) = dr(x ,y-1)cr(y ,z- l )dr(X-I XZ-1).

Hence,

(ll(k~ *~ g)(., y)l]2) 2

feS(A)

2

x ~(xz-', zy-')A(~)-' dz dx H I

= s ~ ) f f "x-'(k(x'z))"z-'(dz'y))~(x'y-l)

2 X f f ( y , z - l ) o ( x - l , x z - 1 ) A ( Z ) -1 dz dx

H I

= sup / /cb:_,(k(x,z))o%_l(~(z,y))cr(y,z -1) feS(A)

2

X O ' ( x - l t xz - l )m(Z ) -1 dz Hf

dx

(11 k~ ,~ ~(., y)ll2) 2 ~ ( l l k ~ l l ~ ) 2 ( l l ~ ( ., y)ll2) 2,

where ~(z, y) = (xz-i (%~(z, y))o(y, z - l ) . Note that ~(., y) C L2(G, A)impliesthat L~(G, A) and that II~(', y)l12 = l id' , y)ll~,2. Hence, I[k~,~C]12,oo = O'(.)-,(~(', y)) E 2

esssupv( II(k~*~{)( ", Y)II2A(y) -1/2) <_ IIk~llggi-~esssupylld. , y)ll2 A(y)-'/2 =

300 RONGHU1 JI AND LARRY B. SCHWEITZER

[I k [I I I 112, , SO the action is continuous. From the fact that the multiplication

in B~(A) (Definition 3.2) is associative, it is easy to see that (k a ,~ l~)~ ,~ ~ = , r2'~176 r":- A). [] k~ ,~. (1. , . ~) for k ~ 1 ~ r B~(A), ~ r ~,A,.~,-.,

DEFINITION 3.8.Note that BG(A) is actually contained in L 2 ,~ t~ A) as func-

tions on G x G. (This is the reason for the factor of A-U2 in Definition 3.6.) Let J be the completion of Ba(A) in the weakest topology which dominates both the

r z ' ~ t ~ A). It is obvious that ~ acts on topology on B~(A) and the topology on -A ,~p- ,

the Banach space L~x~(G,A ) by the same formula as 7 on B~(A) (see Lemma 3.5). The action is also isometric and strongly continuous on BG (A) in the topology

L 2 ' ~ ( ~ A). Thus 7 acts isometrically and strongly continuously on induced from A,~,~, the Banach space J.

COROLLARY 3.9. The Banach space J so defined is a dense left Banach ideal of B~(A).

L 2'~tr: A) (see Proof. This follows from the B~(A)-module structure on A,~v,~',

proof of Theorem 3.7), and the fact that the action of B~(A) on A,~,- ' ,~w

coincides with the multiplication in B~(A). []

PROPOSITION 3.10. (a) Suppose that the group ( G, g) has property SRD and that A is commutative with identity. Let J ~ denote the set ofC~-vectorsfor the action 7 on J. Then

J ~ N B~(A)~ = S~(G,A; a).

If moreover, G is discrete, then

B TY;z : S (G,A;

where B~( A ) -F~ denotes the the set of C~-vectors of"/on B~( A )Z. (b) When A = C, then the statement (a) holds for any (G, g) with property RD. Remark. Note that C~(G, A; a) is identified with B~(A)Z in a natural way as

noted in (3.4). The isometry is given by ~p D ~ q5, where ~5(x, y) = ~(xy -1). Proof. By Theorem 2.2, we may assume the group G is unimodular. Let ~ be

the derivation obtained from the action 3' of R on J. Let/3g(k)(x, y) = k(xg, yg) be the action ~ of G on B~(A) from Lemma 3.3. Let k C J ~ fq B~(A)Z. Then k(x, y) = /c (xy-1) , where/e(x) = k(x, e). Since k e J ~ ,

> II P(k)ll2,

= esssup sup ( [ N(C(x)- s dx) I/2 yeG feS(A) \JG


For all positive integers p, it is possible to find a fixed Y for which this is finite. We have,

du. f E S ( A ) JG

By induction,

vc > sup [g(uy)2pllc~,.-,(s J~S(A) dG'

> f (e(~,) - e(y))2pllc~,,_, (~(~))11~ du.

By induction again,

oc > sup ~ e(u)2Pl[~-,(k(~t))l[~4 j du. f E S ( A )

Since this holds for all p, it follows that k r S~(G, A; cr). Note that this did not use the fact that (G, ~) is SRD nor the assumption that A is commutative.

~g Conversely, assume that ~ E 5e(G, A;cr) and (G,g) has property SRD. Let

�9 y) = ~(xy-'). Then

~'~( i~)( ~, y ) = i , ( e ( x ) - e(y))p~(.~:, y),

and, using the fact that G is unimodular (Theorem 2.2),

116P(@)112,~ = e s s s u p

<

sup (jG(~(x)- ~(y))2Pll(%_t(~(x,y))l[2sdx) I/: yeS(A)

(/ d,) ~,J fES (A)

(~ ( ) fl ( ())112 )~/2 esssup sup [u2P.~u_,~u.~.jdu Y f e S ( A ) \ G

1/2

= Ill~ltl ,~.

Here Illplllp is the seminorm on S~(G, A; cr) defined in (1.2). Therefore, 6P�9 E L2'~(G, A). Now let us use the fact that (G, g) is SRD and that A is commutative. Let P(A) be the set of pure states on A. If ~ C L2(G, A), then by the proof of Lemma 3.5, we have

_< sup [ f (~(x)-~(y))Pf(o,~_~(~(xy-t))) f eP(A) aClaC

302 RONGHU1 JI AND LARRY B. SCHWEITZER

2 x a(x- l ,xy- l ) f (~(y) )dy dx

_< sup [ f (e(x)-g(s-ix))P(ax- '(~(s)) f6P(A) J G II J G

X O-(X -1 , 8)(~(8-1X)) ds 2 dx H I

qO 2 ~ 2 _< sup [] P[[B---~-~[[ [[2 --< r f6P(A)

for some constants d > 0, c > 0, by Proposition 2.3, where ~p(X,y) = g(xy-1)P]f(ax_,(7~(xy-a)))l. Hence ~ E J~ n C;(G,A). Note that since q~ E 5'2~(G, A, ~r), it is clear that ~p E BS2e(G, C) for any pure state f E P ( d ) , which defines a one-dimensional representation of A.

If G is discrete, then for any k E B ~ ( A ~ ~, we have

oo > l16~(k)ll -- sup{ll 5P(k) *,~ ~112 I , c e L2(G,A), II~li2 -~ 1}

= sup sup ( ~ ia(g(x)-g(y))Paz- , (k(x ,y)) IlgN2=l yeS(A)

2 ) 1/2

• { (y )a(x - l , xy - l )dy HS dx

>_ sup ( / [Ig(x)P%:-i(k(x,e))N2Hs dx) 1/2 fES(A) \JG

: sup ( i ' ' /2 : IIIDIII,, ZES(A) \aG

by evaluating at ~ = ~5,, the step function at e = 1G. It follows that ~: E Se2(O, A; ~r). The converse follows from the same argument as for a general group G above.

For (b), everything reduces to a trivial action so that one needs only the property RD rather than the property SRD.

This completes the proof of Proposition 3.10. []

LEMMA 3.11. Let I be a dense left Banach ideal in a Banach algebra B. Moreover, assume that IIb~lls < [IbNBN~lb for b E B, ~ E I. Assume that ~ acts strongly continuously on I, and let I ~176 denote the Frdchet algebra of C ~~ Then I ~176 satisfies the DSC in B.

Proof Let r be the infinitesimal generator for the action of 1~ on [. Let ~o, ~b E [. Topologize I ~176 by the seminorms

I1~ollo = II~llB,


1 i1~_]~11,, k : 1 2,3, II~ll~ = II~%IIB + (k- 1)-------~, ' "

We have

1 1 1 5j II~*Ollk _< ~ IlI6/~II-T, IIU~IIB+ ~ 7.,1[6/~11B7.,11 ~11, i + j = k " i + j = k - I ' "

1 5~ 1 Z 7.,11 ~IIBf.,II~JOIIB

i+j=~ �9

1 ~5 i 1 + ~ ~11 ~IIB(j -1)!ll6J-'011/ i + j = k , j r "

~116%11B[1~110

1 t5 i 1 ),ll~J_l~lli) i+j=k,j:~O " " (j 1 .

i + j = k i + j = k

This proves the Lemma. []

Proof of Theorem 2.6. By Lemma 3.11 and Corollary 3.9, J ~ satisfies the DSC in B~(A). By Proposition 3.10, J~ A C~(G, A; c~) = Sez(G, A; a)if(G, 6)is SRD.

Hence the seminorms on joo and the norm on B~(A) agree with the seminorms on S~(G, A; or) and the norm on C~(G, A; ~), respectively. So 5)e(G, A; or) satisfies the DSC in C~(G, A; c~).

If G is discrete,

,g (Proposition 3.10) implies more directly that 52(G,A;cr) satisfies DSC in

B~;( A) and, hence, in C~(G, A; or). []

A c k n o w l e d g e m e n t s

It is our pleasure to thank Guoliang Yu, Jonathan Rosenberg, and Theodore Palmer for helpful discussions, and to thank Alain Valette for his encouragement and for sending us a copy of his joint paper with Paul Jollisaint [JoV], in which the unimodularity of an RD group was in concern. Finally, we would like to thank Pierre Julg for pointing out a major mistake in the earlier version of this paper, and we are grateful to him for subsequent helpful discussions.


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Spectral invariance of smooth crossed products, and rapid decay locally compact groups

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