AUTOMATIC CONTINUITY OF ORTHOGONALITY

PRESERVERS ON A NON-COMMUTATIVE Lp(τ) SPACE

TIMUR OIKHBERG AND ANTONIO M. PERALTA

Abstract. Elements a and b of a non-commutative Lp(τ) space asso-ciated to a von Neumann algebra N , equipped with a normal semifinitefaithful trace τ , are called orthogonal if l(a)l(b) = r(a)r(b) = 0, wherel(x) and r(x) denote the left and right support projections of x. A linearmap T from Lp(N, τ) to a normed space Y is said to be orthogonality-to-p-orthogonality preserving if ‖T (a) +T (b)‖p = ‖a‖p + ‖b‖p whenevera and b are orthogonal. In this paper, we prove that an orthogonality-to-p-orthogonality preserving linear bijection from Lp(N, τ) (1 ≤ p < ∞,p 6= 2) to a Banach space X is automatically continuous, whenever Nis a separably acting von Neumann algebra. If N is a semifinite factornot of type I2, we establish that every orthogonality-to-p-orthogonalitypreserving linear mapping T : Lp(N, τ) → X is continuous, and in-vertible whenever T 6= 0. Furthermore, there exists a positive con-stant C(p) (1 ≤ p < ∞, p 6= 2) so that ‖T‖‖T−1‖ ≤ C(p)2, for ev-ery non-zero orthogonality-to-p-orthogonality preserving linear mappingT : Lp(N, τ) → X. For p = 1, this inequality holds with C(p) = 1 –that is, T is a multiple of an isometry.

1. Introduction: orthogonality and p-orthogonality

Let X be a normed space, and 1 ≤ p ≤ ∞. Elements x, y in X aresaid to be p-orthogonal (x ⊥p y) if ‖x + y‖p = ‖x‖p + ‖y‖p. For p = ∞,the term M -orthogonality has been historically used interchangeably with∞-orthogonality. Generalizing this notion, we say that x and y are semi-M -orthogonal (x ⊥SM y) if ‖x+ y‖ ≥ max‖x‖, ‖y‖.

The notion of p-orthogonality arises naturally in the settings when eitheralgebraic structure or order are present. Such situations occur in C∗-algebrasand Lp spaces (commutative or non-commutative), as described below.

Let A be a C∗-algebra. Two elements a, b in A are said to be (alge-braically) orthogonal (written a ⊥ b) if ab∗ = b∗a = 0. It is well knownthat orthogonal elements in A are (geometrically) M -orthogonal, while thereciprocal statement is not, in general, true. Many efforts have been made

The first author was supported by the COR grant of the UC system, and by a travelgrant of Simons Foundation. Second author partially supported by the Spanish Ministryof Science and Innovation, D.G.I. project no. MTM2011-23843, and Junta de Andalucıagrants FQM0199 and FQM3737.

2 T. OIKHBERG AND A.M. PERALTA

to study the general form of an orthogonality preserving linear mapping be-tween two C∗-algebras, assuming continuity or not (see e.g. [2], [14], [33],[13], [15], [1], [5], [6], [7], [19], [18], [32], [8], and [21] among others).

Now suppose M is a von Neumann algebra, equipped with a normalsemifinite faithful trace τ , and acting on a Hilbert space H. Following[20] (see also [10, 25]), we say that a closed densely defined (in general,unbounded) operator a is affiliated with M if it commutes with M ′ (thecommutant of M). It follows from von Neumann’s bicommutant theoremthat an element x ∈ L(H) s affiliated with M if and only if x ∈M .

The left and right support projections of a (denoted by l(a) and r(a))are defined as the orthogonal projections onto the closure of the range of a,and the orthogonal complement of the kernel of a. These projections areknown to belong to M . Furthermore, if ea = a for a projection e, thene ≥ l(a). Similarly, if af = a for a projection f , then f ≥ r(a). We saythat two operators a and b, affiliated with M , are orthogonal (a ⊥ b) ifl(a)l(b) = r(a)r(b) = 0. Equivalently, a∗b = ab∗ = 0 [26, Section 1].

We denote by S the set of all linear combinations of positive elementsa ∈ M , satisfying τ(supp(a)) < ∞, where supp(a) denotes the supportprojection of a. It is known that S is a ∗-subalgebra of M which is weak∗-dense in M . Further, for any 1 ≤ p <∞ and x ∈ S the support projectionof |x|p has finite trace (and so τ(|x|p) <∞), where |x| = (x∗x)1/2. Following

standard notation, we define ‖x‖p = (τ(|x|p))1/p (x ∈ S). The space Lp(τ)is defined as the completion of S in the norm ‖ · ‖p. The trace τ extends toa linear functional on S, which will be still denoted by τ . In this case,

|τ(x)| ≤ ‖x‖1 (x ∈ S).

Therefore τ extends to a continuous functional on L1(τ).

If a von Neumann algebra M is acting on a Hilbert space H, then everyelement in Lp(τ) can be regarded as closed densely defined operators on Haffiliated with M . For every closed densely defined self-adjoint operator a ona Hilbert space H there exists a unique spectral measure ea : B(R)→ L(H)(that is, a measure which takes values in the set of all projections in L(H),sends disjoint sets to orthogonal projections and is σ-additive with respectto the strong operator topology) such that a admits a spectral resolution

a =

∫Rλdea(λ),

where B(R) is the Borel σ-algebra of R. An affiliated operator x is said to be

τ -measurable or measurable if there exists λ > 0 such that τ(e|x|(λ,∞)) <∞. The space, L0(M, τ), of all measurable operators on M admits a struc-ture of metrizable topological complex ∗-algebra with unit element 1 andthe von Neumann algebra M is a dense ∗-subalgebra of L0(M, τ). Thetrace τ can be extended to a positive tracial functional on the positive part,

AUTOMATIC CONTINUITY 3

L0(M, τ)+, of L0(M, τ), again denoted by τ , satisfying:

Lp(τ) =x ∈ L0(M, τ) : ‖x‖p <∞

.

Furthermore, Lp(τ) is an M -bimodule, and ‖ax‖p ≤ ‖a‖∞‖x‖p, ‖xa‖p ≤‖a‖∞‖x‖p, ‖x‖p = ‖x∗‖p = ‖|x|‖p for each a ∈ M , x ∈ Lp(τ), where ‖ · ‖∞denotes the C∗-norm on M (cf. [30, §IX.2, Theorem IX.2.13], [35] and [27]).Following standard notation we set L∞(τ) = M equipped with the originalC∗-norm.

By [26, Fact 1.3], the elements a, b ∈ Lp(τ) are orthogonal if and only if

‖a+ b‖pp = ‖a− b‖pp = ‖a‖pp + ‖b‖pp,

that is, if and only if a is p-orthogonal to both b and −b.The reader is referred to [23, 25, 26] for more information on measurable

operators affiliated with a von Neumann algebra, and on non-commutativeLp spaces.

An operator T : Lp(τ) → X is called orthogonality-to-p-orthogonalitypreserving (O-p-O preserving, for short) if T (x) ⊥p T (y) whenever x ⊥ y.We are interested in the structure of such maps, with particular emphasison automatic continuity. In Section 2, we deal with the case of operators onLp(M, τ), where M is a semifinite factor not of type I2. Theorem 2.1 showsthat, for p ∈ [1, 2)∪ (2,∞), such a T must be continuous, and, if T 6= 0, alsoinvertible. Moreover, there exists a positive constant C(p), depending onlyon p, such that the inequality ‖T‖‖T−1‖ ≤ C(p)2 holds for every non-zeroorthogonality-to-p-orthogonality preserving linear mapping T : Lp(τ) →X. For p = 1, we can take C(p) = 1 (Corollary 2.4). Consequently, anyorthogonality-to-1-orthogonality preserving linear mapping T : L1(τ) → Xis a multiple of an isometry.

Section 3 deals with the case when (M, τ) is a separably acting von Neu-mann algebra, but not necessarily a factor. Theorem 3.6 proves that everyorthogonality-to-p-orthogonality preserving linear bijection from Lp(M, τ)to a Banach space is automatically continuous. The proof is obtained bycombining the results of Section 2 with Theorem 3.1, establishing the auto-matic continuity of certain disjointness to semi-M -orthogonality preservingmaps.

Note that, when M coincides with L(H) (the algebra of all boundedoperators on a complex Hilbert space H), and τ is the usual trace on L(H),the associated Lp-space Lp(M, τ) is the Schatten class Sp(H). In a recentpaper (see [22]), D. Puglisi and the authors of this note establish an explicitdescription of orthogonality preserving linear bijections between Schattenideals Sp(H) (1 ≤ p < ∞). The proof relies on a previous result whichassures the automatic continuity of every orthogonality-to-p-orthogonalitypreserving linear bijection from Lp(M, τ) to a Banach space when 1 ≤ p <∞, and M is either an abelian or a discrete von Neumann algebra. The

4 T. OIKHBERG AND A.M. PERALTA

just quoted paper may be considered as the first step to solving the problemtreated in this note.

2. Orthogonality preservers on factors

In this section, we establish automatic continuity results for orthogonality-to-p-orthogonality preserving operators from Lp(τ) to X, where X is anarbitrary Banach space, and τ is a semifinite normal faithful trace on afactor N . The following definitions will be used throughout the section. Fora von Neumann algebra N , we denote by P(N) the set of all projectionsin N . If τ is a normal faithful semifinite trace on N , we denote by Pτ (N)the set of all projections with finite trace. Henceforth, projections in N areassumed to be non-zero. For 1 ≤ p < ∞ and a linear orthogonality-to-p-orthogonality preserving map T : Lp(τ)→ X, define

α(T ) = infq∈Pτ (N)

‖T (q)‖τ(q)1/p

,

Theorem 2.1. For p ∈ [1,∞)\2, there exists a positive constant C =C(p) with the following property: Suppose N is a von Neumann factor, notof type I2, equipped with a normal faithful semifinite trace τ , X is a Banachspace, and T : Lp(τ) → X is an orthogonality-to-p-orthogonality preservinglinear map. Then the inequality

C−1α(T )‖φ‖p ≤ ‖T (φ)‖ ≤ Cα(T )‖φ‖pholds for every φ ∈ Lp(τ). Consequently, if T 6= 0, then ‖T‖‖T−1‖ ≤ C(p)2.

Remark 2.2. No such constant C exists for p = 2, as can be illustrated bythe following example. Suppose a is a strictly positive n× n matrix. Thenthe map T : S2(`2n) → S2(`2n) : φ 7→ aφ is orthogonality-to-2-orthogonalitypreserving. Indeed, if φ ⊥ ψ, then

‖T (φ+ ψ)‖2 = tr((φ+ ψ)∗a2(φ+ ψ)

)= tr

(a2(φ+ ψ)(φ∗ + ψ∗)

)= tr

(a2φφ∗

)+ tr

(a2ψψ∗

)= ‖T (φ)‖2 + ‖T (ψ)‖2.

On the other hand, ‖T‖‖T−1‖ = ‖a‖‖a−1‖ can be arbitrarily large.

Throughout the proof, we rely on the theory of semifinite factors andtheir traces, as explained, for instance, in [17, Sections 8.4-5], or [29, Sec-tion V.2]. It is known that any semifinite von Neumann factor N admitsa normal faithful semifinite trace τ . This trace is unique, up to a multi-plicative constant ([29, Corollary V.2.32]). N is finite if and only if τ isfinite. Furthermore, a projection is finite (resp. infinite) iff its trace is finite(resp. infinite) ([17, Proposition 8.5.2]). Two finite projections in a factorare equivalent iff their traces coincide [17, Theorem 8.4.3]. If e and f areprojections in a factor, such that e is infinite, and f is σ-finite, then f ≺ e(see [17, Theorem 6.3.4.] or [29, Proposition V.1.39]). Recall that a pro-jection q in a von Neumann algebra N is said to be σ-finite (or countably

AUTOMATIC CONTINUITY 5

decomposable) when each orthogonal family of non-zero subprojections of qin N is at most countable.

Remark 2.3. In the proofs, we view a factor N as equipped with a faith-ful normal semifinite trace τ . The particular choice of the trace does notinfluence our calculations. Indeed, suppose τ ′ is another normal faithfulsemifinite trace on N . By the above, there exists a constant c ∈ (0,∞) sothat τ ′ = cτ . Then the spaces Lp(τ) and Lp(τ ′) coincide as sets. Further-more, for any φ ∈ Lp(τ),

‖φ‖τ ′,p =(τ ′(|φ|p)

)1/p= c1/p

(τ(|φ|p)

)1/p= c1/p‖φ‖τ,p,

and consequently, c1/p‖T (φ)‖/‖φ‖τ ′,p = ‖T (φ)‖/‖φ‖τ,p. Similarly, one checksthat

c1/pατ ′(T ) = c1/p inf‖T (q)‖/τ ′(q)1/p : q ∈ Pτ (N)

= inf‖T (q)‖/τ(q)1/p : q ∈ Pτ (N) = ατ (T ).

Thus, it suffices to prove Theorem 2.1 for one trace on N .

In view of the above Remark, we may assume that every finite trace isunital.

The proof of Theorem 2.1 proceeds in two stages. First, we deal with finitefactors (Subsection 2.1), and then tackle the general case (Subsection 2.2).Before proceeding, we wish to show that, for p = 1, T as in the statementof Theorem 2.1 is a multiple of an isometry.

Corollary 2.4. Suppose N is a von Neumann factor, not of type I2, equippedwith a normal faithful semifinite trace τ , X is a Banach space, and T :L1(τ) → X is an orthogonality-to-1-orthogonality preserving linear map-ping. Then T is a multiple of an isometry.

First we establish Corollary 2.4 in the separably acting case. Recall thata von Neumann algebra is called separably acting if it has a normal faithfulrepresentation on a separable Hilbert space. This happens precisely whenits predual is separable (see e.g. [34, Lemma 1.8]).

Lemma 2.5. In the notation of Corollary 2.4, suppose N is a separablyacting finite factor. Then T is a multiple of an isometry.

Proof. A faithful normal trace τ on a finite factor N is necessarily finite. ByRemark 2.3, we can assume τ(1) = 1.

We are only interested in the case of T 6= 0. Theorem 2.1 shows thatthere exist C,D > 0 so that C‖φ‖1 ≤ ‖T (φ)‖ ≤ D‖φ‖1 for any φ ∈ L1(τ).Without loss of generality, we assume ‖T (1)‖ = 1. We shall show that‖T (φ)‖ = ‖φ‖1 for any φ.

Fix a unitary u ∈ N . For a projection q ∈ N , define Φu(q) = ‖T (uq)‖.Clearly, Φu(q1 + q2) = Φu(q1) + Φu(q2) whenever q1q2 = 0. By [24, Theorem6.1], there exists a positive au ∈ N∗ ∼= L1(τ) (cf. [29, Theorem V.2.18]), sothat Φu(q) = τ(auq) for any projection q. Now suppose φ is an algebraic

6 T. OIKHBERG AND A.M. PERALTA

element in N , i.e. φ =∑n

i=1 ciqi, where q1, . . . , qn are mutually orthogonalprojections, and c1, . . . , cn are scalars. Then ‖T (uφ)‖ =

∑ni=1 |ci|‖T (uqi)‖ =

τ(au|φ|). For each normal element φ ∈ N , we can find a sequence (φn) ofalgebraic elements in N converging to φ in the C∗-norm, ‖ · ‖∞, of N . It isclear that

‖|φ− φn|‖2∞ = ‖|φ− φn|2‖∞ = ‖(φ− φn)∗(φ− φn)‖∞ = ‖φ− φn‖2∞ → 0,

and thus ‖φ − φn‖1 = τ(|φ − φn|) ≤ ‖|φ − φn|‖∞ → 0. Since, in this case,|τ(au(φ−φn))| ≤ ‖au(φ−φn)‖1 ≤ ‖au‖1‖φ−φn‖∞ → 0, it follows from thecontinuity of T that ‖T (uφ)‖ = τ(auφ), for every normal element φ ∈ N .

Let ϕ denote the mapping defined by the assignment x 7→ τ(aux) (x ∈M).It follows from the triangle inequality that

ϕ(|φ1 + φ2|) = τ(au|φ1 + φ2|) = ‖T (u(φ1 + φ2))‖≤ ‖T (uφ1)‖+ ‖T (uφ2)‖ = τ(au|φ1|) + τ(au|φ2|) = ϕ(|φ1|) + ϕ(|φ2|),

for every φ1, φ2 in Nsa. By [31, Corollary 3] and the uniqueness (up topositive scalar multiples) of the trace, ϕ(·) = τ(au·) is a positive scalarmultiple of τ . Thus, au = γu1, for some γu > 0. However, 1 = ‖T (1)‖ =‖T (uu∗)‖ = γu, hence γu = 1 for every u. Given an arbitrary element φ ∈ N ,representing φ = u|φ|, where u ∈ N is a unitary, and |φ| ≥ 0, we concludethat ‖T (φ)‖ = ‖φ‖1 for any φ ∈ N .

To handle the general case of T : L1(τ) → X, suppose π : Mn → N(n ∈ N) is a unital representation, while U, V ∈ U(N) (the unitary group ofN). Define the map SU,V,π : Mn → N : a 7→ Uπ(a)V . If trn is the canonicalunital trace on Mn, and τ is the (unique) normal finite faithful unital traceon N , then SU,V,π defines an orthogonality preserving isometry from L1(trn)to L1(τ).

Lemma 2.6. Suppose τ is a normal finite faithful unital trace on a type II1

factor N , and φ ∈ L1(τ). Then for any ε > 0 there exist n ∈ 3, 4, . . ., aunital representation π : Mn → N , unitaries U, V ∈ N , and ψ ∈ ran (SU,V,π),so that ‖φ−ψ‖1 < ε. Furthermore, if φ is normal, we can take U = V = 1.

Proof. Without loss of generality, ‖φ‖1 = 1, and ε < 1 (and τ(1) = 1). Writeφ = u|φ|, where u is a unitary. Find m ∈ N, c1, . . . , cm ≥ 0, and mutuallyorthogonal non-zero projections q1, . . . , qm ∈ N , so that (i)

∑mi=1 qi = 1,

(ii)

∥∥∥∥∥|φ| −m∑i=1

ciqi

∥∥∥∥∥1

< ε/2, and (iii)

∥∥∥∥∥m∑i=1

ciqi

∥∥∥∥∥1

=m∑i=1

|ci| = 1.

Now let L =⌈

2ε

⌉. SetK0 = 0. For 1 ≤ i ≤ m, letKi = bLτ(qi)c. For 0 ≤ i ≤

m, set Ji =∑

s≤iKs (then J0 = 0, and Jm =∑m

i=1Ki ≤ L). Find mutually

orthogonal projections rj (1 ≤ j ≤ L), of trace 1/L each, so that rj ≤ qifor Ji−1 + 1 ≤ j ≤ Ji. The projections rj are equivalent, and

∑Lj=1 rj = 1

(compare [17, Theorem 8.4.3, Proposition 8.5.3]). Therefore, there exists aunital representation π : ML → N so that π(Ejj) = rj for 1 ≤ j ≤ L (here

AUTOMATIC CONTINUITY 7

Eij are the matrix units in ML). Then ψ = u∑m

i=1 ci∑Ji

j=Ji−1+1 rj belongs

to the range of Su,1,π. Furthermore, since Ki = bLτ(qi)c,

q′i = qi −Ji∑

j=Ji−1+1

rj

are projections of trace less than 1/L. Therefore,∥∥∥∥∥ψ − um∑i=1

ciqi

∥∥∥∥∥1

=

∥∥∥∥∥um∑i=1

ciq′i

∥∥∥∥∥1

=

m∑i=1

ciτ(q′i) <

m∑i=1

ciL−1 <

ε

2.

By the triangle inequality, ‖φ− ψ‖1 < ε.If φ is normal, then there exist mutually orthogonal projections qi ∈ N ,

adding up to 1, and scalars ci ∈ C, so that ‖φ−∑m

i=1 ciqi‖1 < ε/2. Define the

projections rj as above, and let ψ =∑m

i=1 ci∑Ji

j=Ji−1+1 rj . Then ‖φ−ψ‖1 <ε, and ψ belongs to the range of S1,1,π, for an appropriate representationπ.

Proof of Corollary 2.4. By Theorem 2.1, T is continuous. Without loss ofgenerality, we assume ‖T (1)‖ = 1. Lemma 2.5 proves our corollary forN = Mn, with n ∈ 3, 4, . . .. Now suppose N is a type II1 factor. ByLemma 2.5, for any n ≥ 3, unitaries U, V ∈ N, and a unital representationπ : Mn → N , there exists a constant γU,V,π ∈ (0,∞) so that ‖T (φ)‖ =γU,V,π‖φ‖1 for any φ ∈ ran (SU,V,π). We shall show that γU,V,π = 1. Notefirst that γ1,1,π = ‖T (1)‖ = 1, for any unital representation π. Similarly,for general U and V , γU,V,π = ‖T (UV )‖. However, UV is a unitary, hence,by the “furthermore” statement of Lemma 2.6, for every ε > 0 there existsψ ∈ ran (S1,1,ρ) (ρ is a representation), so that ‖UV − ψ‖1 < ε. As notedabove, ‖T (ψ)‖ = ‖ψ‖1. As ε can be arbitrarily small, we conclude thatγU,V,π = ‖T (UV )‖ = 1. Thus, T is an isometry on

⋃U,V,π ran (SU,V,π). By

Lemma 2.6, the latter set is dense in L1(τ). By the continuity of T , it is anisometry.

Finally, suppose the trace τ is merely semifinite. If q is a projection offinite trace, then qNq is a factor, whose predual can be identified with qN∗q.By the reasoning above, there exists γq > 0 so that ‖T (φ)‖ = γq‖φ‖1 forany φ ∈ N∗ satisfying qφq = φ. Note that γq = γr if q and r are finite traceprojections. Indeed, q ∨ r is a finite projection [17, Theorem 6.3.8], henceγq = γq∨r = γr. Thus, there exists γ > 0 so that ‖T (φ)‖ = γ‖φ‖1 for anyφ ∈ N∗ for which there exists a finite projection q satisfying qφq = φ. By thecontinuity of T , the equality ‖T (φ)‖ = γ‖φ‖1 must hold for any φ ∈ N∗.

2.1. The proof of Theorem 2.1 for finite factors. Henceforth, if N is avon Neumann algebra, we denote by N+ (Nsa, Nnor) the set of positive (re-spectively, self-adjoint or normal) elements of N . The definition of Lp(τ)+,Lp(τ)sa, and Lp(τ)nor is similar.

8 T. OIKHBERG AND A.M. PERALTA

Lemma 2.7. Suppose p ∈ [1,∞)\2, and let N2 be the set of all non-zeropositive 2× 2 matrices a such that

(2.1)(tr(a|x+ y|p)

)1/p ≤ (tr(a|x|p))1/p +(tr(a|y|p)

)1/pfor any x, y ∈ (M2)sa. Then every element in N2 is invertible. Moreover,there exists a constant C1 = C1(p) > 0, depending only on p, such that‖a‖‖a−1‖ < C1, for every a in N2.

Proof. Let us notice that N2 is closed and R+N2 = N2. Let a be an elementin N2. Suppose, for the sake of contradiction, that a is not invertible. Wemay assume that

tr(a|x+ y|p)1/p ≤ tr(a|x|p)1/p + tr(a|y|p)1/p

for any x, y ∈ (M2)sa, and a =

(1 00 0

). In other words,⟨

|x+ y|p(

10

),

(10

)⟩1/p

≤⟨|x|p

(10

),

(10

)⟩1/p

+

⟨|y|p

(10

),

(10

)⟩1/p

for any x, y ∈ (M2)sa. Considering uxu∗ and uyu∗ instead of x and y (whereu is any 2× 2 unitary), we conclude that

(2.2) 〈|x+ y|pξ, ξ〉1/p ≤ 〈|x|pξ, ξ〉1/p + 〈|y|pξ, ξ〉1/p .for any ξ ∈ `22. We shall show that, for p ∈ [1,∞)\2, there exist x, y ∈(M2)sa and ξ ∈ `22 failing (2.2).

To this end, consider u =

(−1 11 −1

)and v =

(2 00 0

). Then |u| = 2η⊗η

and v = 2ζ ⊗ ζ, where η = 1√2

(1−1

), and ζ =

(10

). Furthermore, |u+ v| =

√2I. Let ξ =

(01

). Then 〈|u+ v|pξ, ξ〉1/p =

√2, and 〈|v|pξ, ξ〉1/p = 0.

Furthermore,〈|u|pξ, ξ〉 = 2p|〈ξ, η〉|2 = 2p−1.

For p < 2, set x = u and y = v. Then√

2 = 〈|x+ y|pξ, ξ〉1/p > 〈|x|pξ, ξ〉1/p + 〈|y|pξ, ξ〉1/p = 21−1/p,

witnessing the failure of (2.2).For p > 2, set x = u+ v, and y = −v. Then x+ y = u, and

21−1/p = 〈|x+ y|pξ, ξ〉1/p > 〈|x|pξ, ξ〉1/p + 〈|y|pξ, ξ〉1/p =√

2,

which shows that (2.2) fails and gives the desired contradiction.We have shown that N2 is a closed subset of the set of invertible elements

of M2. We shall finally show the existence of the constant C1. We observethat the set ‖a‖‖a−1‖ : a ∈ N2 is bounded if and only if the set ‖a−1‖ :a ∈ N2, ‖a‖ = 1 is. Therefore, if no such a constant C1 exists the abovesets are unbounded, so there exists a sequence (an) in N2 with ‖an‖ = 1, forevery natural n, and ‖a−1

n ‖ → ∞. It follows from a compactness argument

AUTOMATIC CONTINUITY 9

together with the upper semi-continuity of the spectrum that there exists asubsequence (aσ(n)) converging to some a ∈ M2 in norm with ‖a‖ = 1 anda not invertible in M2. In particular, a ∈ N2, which is impossible.

Remark 2.8. Let τ be a finite trace on a von Neumann algebra M and leta be a positive element in M . The assignment (x, y) 7→ τ(xay∗) definesa (semi-)positive sesquilinear form on M × M . This form gives rise toa pre-Hilbertian seminorm defined by ‖x‖a,2 = τ(xax∗) (x ∈ M). TheCauchy-Schwarz inequality guarantees that

τ(xay + yax) ≤ 2τ(ax2)12 τ(ay2)

12 ,

for every x, y ∈Msa, equivalently, the inequality(τ(a|x+ y|2)

)1/2 ≤ (τ(a|x|2))1/2

+(τ(a|y|2)

)1/2,

is satisfied for any x, y ∈ Msa. In particular, for p = 2, the inequality (2.1)always holds for every a ≥ 0 in M2. Thus, we have

C1(2) = sup‖a‖‖a−1‖ : 0 ≤ a invertible in M2 =∞.For p = 1, the proof of Lemma 2.7 shows that the function t 7→ |t| is not

operator convex (see e.g. [3, Example V.1.1]). One should also note that ifa non-negative matrix a ∈M2 has the property that

tr(a|x+ y|) ≤ tr(a|x|) + tr(a|y|)for any x, y ∈ (M2)sa, then a is a multiple of the identity (cf. [31, Corollary3]). Thus, any matrix a satisfying the hypothesis of Lemma 2.7 with p = 1must be a multiple of the identity, that is, C1(1) = 1. See also [4, 28] forsome related results. It is also clear from the above proof that C1(p) ≥ 1,for every p ∈ [1,∞)\2.

Let M be a von Neumann algebra equipped with a finite trace τ . Inspiredby the above remark, for each a ∈ M+, x ∈ M, and p ≥ 1, we define

‖x‖a,p := τ(a|x|p)1p . Denote by Mp = Mp(M) the set of all a ∈ M+ for

which ‖ · ‖a,p is a semi-norm on Msa (that is, the triangle inequality issatisfied). We write Mn,p for Mp(Mn).

Lemma 2.9. Let M be a von Neumann algebra equipped with a finite traceτ . For p 6= 2, the set Mp is a closed convex cone, closed under unitaryconjugation – that is, for a unitary u ∈ M and a ∈ M+, a ∈ Mp if, andonly if, uau∗ ∈ Mp. Moreover, M2,p consists of the invertible matricesa ∈ (M2)+, satisfying ‖a‖‖a−1‖ ≤ C1(p).

Proof. The property of Mp being closed is the easiest. To prove that Mp

is a convex cone, note that, for any t > 0, a ∈Mp iff ta ∈Mp. This provesthe homogeneity. To establish the convexity, suppose a, b ∈Mp. Then

‖ · ‖a+b,p =(‖ · ‖pa,p + ‖ · ‖pb,p

) 1p ,

hence ‖ · ‖a+b,p is also a semi-norm. Consequently, Mp is convex.

10 T. OIKHBERG AND A.M. PERALTA

To show thatMp is closed under unitary conjugation, consider a unitaryu ∈M , and a ∈M+. For each x ∈Msa,

‖x‖puau∗,p = τ(uau∗|x|p) = τ(au∗|x|pu) = τ(a|xu|p) = ‖xu‖pa,p,

which proves the desired property.Now consider the 2×2 case. By Lemma 2.7, if a ∈M2,p, then a is invert-

ible and ‖a‖‖a−1‖ ≤ C1(p). Conversely, suppose a is invertible and positivewith ‖a‖‖a−1‖ ≤ C1(p). Write a = ‖a‖ubu∗, where u is a unitary, and

b =

(1 00 λ

), with 1/C1(p) ≤ λ ≤ 1. When C1(p) = 1, a = ‖a‖I2 ∈ M2,p,

otherwise, for θ = (1−λ)C1(p)C1(p)−1 , b = θc + (1 − θ)I2, where c =

(1 00 1/C1(p)

).

As M2,p is closed and convex, we deduce, by Lemma 2.7, that c ∈ M2,p,and hence b ∈M2,p.

Problem 2.10. What is is the value of C1(p) for 2 6= p > 1?

Suppose now that τ is a faithful normal semifinite trace on a von Neumannalgebra N . For 1 ≤ p < ∞ and a linear orthogonality-to-p-orthogonalitypreserving map T : Lp(τ)→ X, define

(2.3) α(T ) = infq∈Pτ (N)

‖T (q)‖τ(q)1/p

and β(T ) = supq∈Pτ (N)

‖T (q)‖τ(q)1/p

.

Note that, for two mutually orthogonal projections q1 and q2,

‖T (q1)‖p + ‖T (q2)‖p = ‖T (q1 + q2)‖p.

Therefore, for any r, q ∈ Pτ (N), satisfying 0 < r < q, either

‖T (q − r)‖τ(q − r)1/p

≤ ‖T (r)‖τ(r)1/p

or‖T (q − r)‖τ(q − r)1/p

≥ ‖T (r)‖τ(r)1/p

,

thus either

(2.4)‖T (q − r)‖τ(q − r)1/p

≤ ‖T (q)‖τ(q)1/p

≤ ‖T (r)‖τ(r)1/p

or‖T (q − r)‖τ(q − r)1/p

≥ ‖T (q)‖τ(q)1/p

≥ ‖T (r)‖τ(r)1/p

.

In particular, if N is discrete, then it suffices to take infimum and supremumin (2.3) over rank 1 projections.

Lemma 2.11. Let N be a von Neumann factor of type In (3 ≤ n < ∞)with a normal faithful finite trace τ, X a Banach space and T : Lp(τ)→ Xan orthogonality-to-p-orthogonality preserving linear map (p ∈ [1,∞)\2).Then ‖T (q)‖ ≤ (C1τ(q))1/pα(T ) for any q ∈ P(Mn), where C1 = C1(p) is

the constant given by Lemma 2.7. In other words, β(T ) ≤ C1p

1 α(T ).

Proof. Define φ : P(Mn) → [0,∞) by setting φ(q) = ‖T (q)‖p. Clearly,φ(q1 +q2) = φ(q1)+φ(q2) whenever q1q2 = 0 in P(Mn). By [11], there existsa non-negative a ∈ (Mn)sa such that φ(q) = τ(aq) for any q. We claim that,for any x ∈ (Mn)nor, ‖T (x)‖p = τ(a|x|p). Indeed, write x =

∑ni=1 λiqi,

AUTOMATIC CONTINUITY 11

where λi ∈ C, and qi are mutually orthogonal rank 1 projections. Note that|x| =

∑ni=1 |λi|qi. As T is O-p-O preserving,

‖T (x)‖p =n∑i=1

|λi|p‖T (qi)‖p =n∑i=1

|λi|pτ(aqi) = τ( n∑i=1

|λi|paqi)

= τ(a|x|p

).

We may assume that a 6= 0, otherwise T = 0. By the triangle inequality, wehave

τ(a|x+y|p

)1/p= ‖T (x+y)‖ ≤ ‖T (x)‖+‖T (y)‖ = τ

(a|x|p

)1/p+τ(a|y|p

)1/p,

for every x, y ∈ (Mn)nor. By Lemma 2.7, for each rank 2 projection r ∈ N ,rar is invertible in rNr ∼= M2 and ‖rar‖‖(rar)−1‖ ≤ C1. In particular, ais invertible. Write a =

∑ni=1 αiri, where r1, . . . , rn are rank 1 projections,

and α1 ≥ . . . ≥ αn > 0. We observe that, for each projection q, ‖T (q)‖p =τ(aq) = τ(qaq) with αnτ(q) = αnτ(q1q) ≤ τ(qaq) ≤ α1τ(q). This implies

that α(T ) = α1pn and β(T ) = α

1p

1 . Now let r = r1 + r2. Applying Lemma 2.7to the matrix rar (acting on the 2-dimensional space ran (r)), we obtainβ(T )p/α(T )p = α1/αn = ‖rar‖‖(rar)−1‖ ≤ C1. Thus, β(T )p ≤ C1α(T )p,as desired.

Suppose N is a von Neumann algebra. We say that a family of partialisometries (uij)1≤i,j≤m in N is a standard m×m system if u∗ij = uji, uijujk =

uik, and uijuk` = 0 if k 6= j. In other words, there exists a *-representationπ : Mm → N , such that the partial isometries uij are images of matrix units.Note that, in this setting, uii are mutually orthogonal equivalent projections.

Lemma 2.12. Suppose τ is the normal faithful semifinite trace on a semifi-nite von Neumann algebra N , and T : Lp(τ)→ X is a linear orthogonality-to-p-orthogonality preserving map (p ∈ [1,∞)\2). Suppose, furthermore,that (uij)1≤i,j≤3 is a standard 3 × 3 system in N , and φ ∈ Lp(N) satisfies

u11φu11 = φ. Then ‖T (u21φu12)‖ ≤ C1(p)1p ‖T (φ)‖, where C1 = C1(p) is the

constant given by Lemma 2.7.

Proof. Let π : M3 → N be a *-representation taking the matrix unit Eij to

uij . Let φ′ =

3∑i=1

ui1φu1i. Consider the maps π′ : M3 → Lp(τ) : a 7→ π(a)φ′,

and T ′ = T π′. Note that π′ is orthogonality preserving. Indeed, for anyk, ` ∈ 1, 2, 3, we have

π(Ek`)φ′ = uk`

( 3∑i=1

ui1φu1i

)= uk1φu1` =

( 3∑i=1

ui1φu1i

)uk` = φ′π(Ek`).

Consequently, π(a) commutes with φ′, for every a ∈ M3. If a, b ∈ M3

are mutually orthogonal, then the same is true for π(a) and π(b), and forπ′(a) = π(a)φ′ and π′(b) = π(b)φ′.

Thus, T ′ is orthogonality-to-p-orthogonality preserving. By Lemma 2.11,

‖T ′(E22)‖ ≤ C1(p)1p ‖T ′(E11)‖, because τ(u22) = τ(u11). To complete the

12 T. OIKHBERG AND A.M. PERALTA

proof, recall that T ′(E22) = T (u21φu12), and T ′(E11) = T (u11φu11) = T (φ).

Lemma 2.13. Suppose τ is the normal faithful semifinite trace on a semifi-nite von Neumann factor N , and T : Lp(τ) → X is a linear orthogonality-to-p-orthogonality preserving map (p ∈ [1,∞)\2). Suppose, furthermore,that φ ∈ Lp(N), and there exists a partial isometry u ∈ N such thatuu∗φuu∗ = φ, and one of the following holds:

(1) The trace τ is finite, and τ(uu∗) ≤ τ(1)/4.(2) The trace τ is infinite, and the projections uu∗ and (uu∗∨u∗u)⊥ are

σ-finite and infinite, respectively.

Then ‖T (u∗φu)‖ ≤ C1(p)2p ‖T (φ)‖, where C1 = C1(p) is the constant given

by Lemma 2.7.

Proof. Let q1 = uu∗ and q2 = u∗u be the range and domain projections ofu.

First, handle case (1). By scaling, assume τ(1) = 1. By KaplanskyFormula ([17, Theorem 6.1.7]), τ(q1 ∨ q2) ≤ τ(q1) + τ(q2) ≤ τ(1)/2, henceτ((q1∨q2)⊥

)≥ τ(1)/2. As N is a finite factor, there exists a projection q3 ≤

(q1 ∨ q2)⊥, equivalent to q1 and q2 (cf. [17, Theorem 8.4.3]). Furthermore,τ((q1 ∨ q2)⊥ − q3

)≥ τ(1)/4 ≥ τ(q1), hence, by [17, Theorem 8.4.3], there

exists a projection q4 ≤ (q1 ∨ q2)⊥ − q3, equivalent to q1. Find partialisometries uij (1 ≤ i, j ≤ 4), implementing the equivalence between theprojections qi, so that u∗ij = uji, uii = qi, uijujk = uik, and uijuk` = 0 if

j 6= k and either j or k exceeds 2, and u12 = u. One can construct (uij) bytaking u11 = q1, and u12 = u (note that the initial and terminal projectionsof u are q2 and q1, respectively). Select partial isometries u1j (j = 3, 4) sothat q1 = u1ju

∗1j and qj = u1ju

∗1j . Set uj1 = u∗1j . For i > 1, let uij = ui1u1j .

Then, for s ∈ 1, 2, (uij)i,j∈s,3,4 is a standard 3× 3 system. ApplyingLemma 2.12 to (uij)i,j∈2,3,4, we see that

‖T (u∗φu)‖ = ‖T (u21φu12)‖ = ‖T (u23u31φu13u32)‖ ≤ C1(p)1p ‖T (u31φu13)‖.

An application of the same lemma to (uij)i,j∈1,3,4 yields

‖T (u31φu13)‖ ≤ C1(p)1p ‖T (φ)‖,

which gives the desired statement.Case (2) is dealt with similarly. The only difference lies in the construction

of projections q3 and q4, which are orthogonal to q1, q2, and each other. Toprove their existence, let r = (q1∨q2)⊥. Then rNr is an infinite factor, hence,by [17, Lemma 6.3.3], we can represent r = r3 + r4, where the projectionsr3 and r4 are equivalent to r. Furthermore, the projections q1 and q2 are σ-finite, hence so is q1 ∨ q2 [16, Exercise 5.7.45]. Consequently, q1 ∼ q2 ≺ r3 ∼r4. This allows us to pick the projections q3 ≤ r3 and q4 ≤ r4, equivalent toq1.

AUTOMATIC CONTINUITY 13

Lemma 2.14. Suppose τ is the unital normal faithful finite trace on a fi-nite von Neumann factor N of rank ≥ 3, and T : Lp(τ) → X is a lin-ear orthogonality-to-p-orthogonality preserving map (p ∈ [1,∞)\2). LetC1 = C1(p) be the constant given by Lemma 2.7. When N is of type In(4 ≤ n <∞) or of type II1 we have

‖T (q)‖p ≤ 2C1(p)2τ(q)α(T )p,

for any q ∈ P(N). When N has rank 3, ‖T (q)‖p ≤ 2C1(p)2+13 τ(q)α(T )p, for

every q ∈ P(N).

Proof. Suppose first, that N has rank bigger or equal than 4. We have toshow that, for any non-zero projections q1 and q2, we have

(2.5)‖T (q1)‖p/τ(q1)

‖T (q2)‖p/τ(q2)≤ 2C1(p)2.

By Lemma 2.13,‖T (q1)‖p/τ(q1)

‖T (q2)‖p/τ(q2)≤ C1(p)2,

whenever τ(q1) = τ(q2) ≤ 1/4.Next we establish the validity of (2.5) when τ(q2) ≤ 1/4 and τ(q1) ≥

τ(q2). Let M = dτ(q1)/τ(q2)e ∈ N. Write q1 as a sum of M mutually

orthogonal equivalent projections: q1 =∑M

s=1 q(s)1 , with τ(q

(s)1 ) = τ(q1)/M

for any s. For each s, there exists a projection r(s) ≥ q(s)1 , equivalent to q2 (by

[17, Theorem 8.4.3] and its proof, this happens iff the equality τ(r(s)) = τ(q2)

holds). By Lemma 2.13, applied to q2 and r(s), we have

‖T (q(s)1 )‖ ≤

(‖T (q

(s)1 )‖p + ‖T (r(s) − q(s)

1 )‖p) 1p

=‖T (r(s))‖ ≤ C1(p)2p ‖T (q2)‖.

Thus,

‖T (q1)‖p

τ(q1)≤ 1

(M − 1)τ(q2)

M∑s=1

‖T (r(s))‖p =M

(M − 1)τ(q2)maxs‖T (r(s))‖p

≤ M

(M − 1)τ(q2)C1(p)2‖T (q2)‖p ≤ 2C1(p)2 ‖T (q2)‖p

τ(q2).

Finally, suppose τ(q2) > minτ(q1), 1/4. Find M ∈ N so that τ(q2)/M ≤minτ(q1), 1/4, and write q2 as a sum of mutually orthogonal equivalent

projections q(s)2 (1 ≤ s ≤ M). Note that, for any s, τ(q

(s)2 ) = τ(q2)/M .

Moreover,∑s

‖T (q(s)2 )‖p = ‖T (q2)‖p, hence there exists s so that ‖T (q

(s)2 )‖p

≤ ‖T (q2)‖p/M . For this s,

‖T (q(s)2 )‖p

τ(q(s)2 )

≤ ‖T (q2)‖p

τ(q2).

14 T. OIKHBERG AND A.M. PERALTA

By our choice of M , τ(q(s)2 ) ≤ min1/4, τ(q1). It follows from the first part

of the proof that

‖T (q1)‖p

τ(q1)≤ 2C1(p)2 ‖T (q

(s)2 )‖p

τ(q(s)2 )

≤ 2C1(p)2 ‖T (q2)‖p

τ(q2),

which is what we need.The above reasoning yields the statement of the lemma when there exist

non-zero projections of trace not exceeding 1/4 – in other words, when ourfactor N is of type In (4 ≤ n < ∞) or II1. If N = M3, we use a differentargument. By (2.4) and compactness, there exists a rank one projection q1 sothat 3‖T (q1)‖p = α(T )p. Find rank one projections q2 and q3, orthogonalto q1 and to each other. By Lemma 2.13, ‖T (qi)‖p ≤ C1(p)2‖T (q1)‖p =C1(p)2

3 α(T )p for i ∈ 2, 3. Therefore,

‖T (1)‖p =

3∑i=1

‖T (qi)‖p ≤2C1(p)2 + 1

3α(T )p.

Thus, for any projection r,

‖T (r)‖p

τ(r)≤ ‖T (r)‖p ≤ ‖T (1)‖p ≤ 2C1(p)2 + 1

3α(T )p.

Lemma 2.15. Suppose τ is the normal faithful unital trace on a (finite)factor N of rank bigger or equal than 3, and T : Lp(τ) → X is a lin-ear orthogonality-to-p-orthogonality preserving map (p ∈ [1,∞)\2). Then

‖T (φ)‖ ≤ 96 21p C1(p)

2pα(T )‖φ‖p for any φ ∈ N .

Proof. First prove the inequality ‖T (φ)‖ ≤ 48 21p C1(p)

2pα(T )‖φ‖p for φ ≥ 0.

If N = Mn with n ≥ 3, find mutually orthogonal rank one projectionsq1, . . . , qn so that φ =

∑ni=1 ciqi, with ci’s non-negative. Then ‖φ‖pp =∑n

i=1 cpi τ(qi), and

‖T (φ)‖p =

n∑i=1

cpi ‖T (qi)‖p ≤n∑i=1

cpiβ(T )pτ(qi)

(by Lemma 2.11) ≤ C1(p)α(T )pn∑i=1

cpi τ(qi) = C1(p)α(T )p‖φ‖pp.

If N is a type II1 factor with a normalized trace τ , find a step functionf : [0, ‖φ‖p] → [0, ‖φ‖p], such that, for every λ, 0 ≤ λ ≤ f(λ) ≤ λ + δ,where δ ∈ (0, ‖φ‖p/τ(1)). Then ‖f(φ) − φ‖p ≤ τ(1)‖f(φ) − φ‖ ≤ ‖φ‖p.There exist mutually orthogonal projections q1, . . . , qN , commuting with φ,so that 0 ≤ φ ≤ f(φ) =

∑Ni=1 ciqi, such that

2p‖φ‖pp > ‖f(φ)‖pp =N∑i=1

cpi τ(qi),

AUTOMATIC CONTINUITY 15

where the coefficients ci are positive. Then the operators φi = c−1i qiφqi

are positive contractions in N . It is easy to see that qiNqi is a type II1

factor for every i (it follows, for instance, from [17, Proposition 6.2.6] or [29,

Proposition II.3.10]). By [12], we can write φi = 2∑6

j=1(q(1)ij − q

(2)ij ), where

q(s)ij ∈ P(qiNqi). By Lemma 2.14,

‖T (q(s)ij )‖ ≤ 2

1pC1(p)

2pα(T )τ(q

(s)ij )1/p ≤ 2

1pC1(p)

2pα(T )τ(qi)

1/p,

hence ‖T (φi)‖ ≤ 24 21pC1(p)

2pα(T )τ(qi)

1/p. The φi’s are orthogonal, hence

‖T (φ)‖p =N∑i=1

cpi ‖T (φi)‖p ≤(24)p 2 C1(p)2α(T )p

N∑i=1

cpi τ(qi)

≤(48)p 2 C1(p)2α(T )p‖φ‖pp.

This proves the statement of the lemma for φ ∈ N+.Any φ ∈ Nsa can be written as φ = φ1−φ2 in such a way that φ1, φ2 ∈ N+,

and φ1 ⊥ φ2. Therefore,

‖T (φ)‖p = ‖T (φ1)‖p + ‖T (φ2)‖p

≤(48)p 2 C1(p)2α(T )p(‖φ1‖pp + ‖φ2‖pp) =

(48)p 2 C1(p)2α(T )p‖φ‖pp.

For a general φ, write φ = φ< + iφ=, where φ< = (φ + φ∗)/2 and φ= =(φ− φ∗)/(2i) are self-adjoint. Note that ‖φ<‖p ≤ (‖φ‖p + ‖φ∗‖p)/2 = ‖φ‖p,and similarly, ‖φ=‖p ≤ ‖φ‖p. Then

‖T (φ)‖ ≤ ‖T (φ<)‖+ ‖T (φ=)‖ ≤(48) 2

1p C1(p)

2pα(T )

(‖φ<‖p + ‖φ=‖p

)≤ 96 2

1p C1(p)

2pα(T )‖φ‖p.

Lemma 2.16. Suppose τ is the normal faithful unital trace on a (finite)von Neumann factor N of rank bigger or equal than 3, and T : Lp(τ)→ Xis an unbounded linear orthogonality-to-p-orthogonality preserving map (p ∈[1,∞)\2). Then, for every ε > 0 and A > 1, there exists a projectionq ∈ N and φ ∈ Lp(τ), such that ‖φ‖p ≤ 1, qφq = φ, τ(q) ≤ ε, and‖T (φ)‖ ≥ A.

Proof. We may assume that N is of type II1, otherwise N and Lp(τ) arefinite-dimensional, and hence, no such a mapping T exists. By Lemma 2.15,we can assume that ‖T (ψ)‖ ≤ ‖ψ‖p for any ψ ∈ N . On the other hand, Tis unbounded, hence

sup‖T (φ)‖ : φ ∈ Lp(τ), φ ≥ 0, ‖φ‖p = 1 =∞.For n ∈ N find φ0 ∈ Lp(N)\N so that ‖φ0‖p = 1, φ0 ≥ 0, and ‖T (φ0)‖ > 4n.Using the spectral decomposition, find mutually orthogonal projections qi(1 ≤ i ≤ 2n), such that φ0 =

∑2n

i=1 qiφ0qi. By the triangle inequality,∑2n

i=1 ‖T (qiφ0qi)‖ ≥ 4n, hence ‖T (qiφ0qi)‖ ≥ 2n for some i. This proves thestatement of the lemma for ε = 2−n, and A = 2n.

16 T. OIKHBERG AND A.M. PERALTA

Proof of Theorem 2.1 for finite factors. Let N be a finite von Neumann fac-tor of type In (n ∈ 3, 4, . . .) or II1, and let τ denote the unital normalfaithful finite trace on N . We first show that for every p ∈ [1,∞)\2, eachorthogonality-to-p-orthogonality preserving linear map T from Lp(τ) to aBanach space X is continuous. The statement is clear when N is a type Infactor with 3 ≤ n <∞.

Let us assume that N is of type II1. Suppose, for the sake of contra-diction, that T is not bounded. By Lemma 2.16, there exist projections qn(n ≥ 2), and elements φn ∈ Lp(τ), such that τ(qn) ≤ 2−n, ‖φn‖p ≤ 1, φn =qnφnqn, and ‖T (φn)‖ > 4n. Find mutually orthogonal projections r1, r2, . . .,such that rn ∼ qn for any n. Let (un) denote the partial isometries imple-menting this equivalence (that is, unu

∗n = rn, and u∗nun = qn), ψn = unφnu

∗n,

and ψ =∑∞

n=1 2−nψn. Clearly, for each natural n, rnψnrn = ψn. Fix anarbitrary natural n ≥ 2, since ψ − 2−nψn is orthogonal to 2−nψn, it followsfrom Lemma 2.13 that

‖T (ψ)‖ ≥ 2−n‖T (ψn)‖ ≥ 2−nC1(p)− 2p ‖T (φn)‖ > C1(p)

− 2p 2n

(for every n ≥ 2), which is impossible.We need to show the existence of a positive constant C = C(p), depending

only on p, such that the inequality

C−1 α(T ) ‖φ‖p ≤ ‖T (φ)‖ ≤ Cα(T )‖φ‖p,

holds for every φ ∈ Lp(τ). Taking C = C(p) = 96 21p C1(p)

2p , the right

inequality follows directly from Lemma 2.15, the density of N in Lp(τ), andthe continuity of T (established in the previous paragraph). Clearly, T = 0if, and only if, α(T ) = 0.

To prove the left inequality, consider first φ ∈ Lp(τ)+, of norm 1. Fixε > 0, and find mutually orthogonal projections q1, . . . , qn, and positivenumbers c1, . . . , cn, such that ‖φ −

∑ni=1 ciqi‖p < ε. By scaling, we can

assume that ‖∑n

i=1 ciqi‖pp =

∑ni=1 c

pi τ(qi) = 1. As T is orthogonality-to-p-

orthogonality preserving, we have∥∥∥∥∥T(n∑i=1

ciqi

)∥∥∥∥∥p

=

n∑i=1

cpi ‖T (qi)‖p ≥n∑i=1

cpiα(T )τ(qi) = α(T ),

hence ‖T (φ)‖ ≥ α(T ) − ‖T‖ε. As ε > 0 is arbitrary, we conclude that‖T (φ)‖ ≥ α(T ).

For a unitary u ∈ N , let Lu be the operator of left multiplication by u– that is, for ψ ∈ Lp(τ), Luψ = uψ. Let Tu = T Lu. Clearly, Lu is asurjective isometry on Lp(τ), hence ‖T‖ = ‖Tu‖. Moreover, Lu preservesorthogonality, and hence Tu is orthogonality-to-p-orthogonality preserving.Next, we claim that α(Tu) ≥ α(T ) C−1. Indeed, as noted above,

α(T ) ≤ ‖T‖ = ‖Tu‖ ≤ Cα(Tu),

which gives the desired inequality.

AUTOMATIC CONTINUITY 17

Now consider the case of a general norm-one element φ ∈ Lp(τ). Usepolar decomposition to write φ = u|φ|, where u is a unitary, and |φ| ≥ 0.Applying the above argument to Tu we get

‖T (φ)‖ = ‖Tu(|φ|)‖ ≥ α(Tu) ≥ C−1α(T ),

completing the proof.

2.2. Proof of Theorem 2.1 in the semifinite case. Let τ denote afaithful normal semifinite trace on N and let a be a positive element in Nwith τ(a) <∞. It is not hard to see that the range or support projection ofa is σ-finite. Indeed, let r denote the range projection of a and suppose that(qj) is a family of mutually orthogonal projections such that qj ≤ r, for every

j. Clearly, a(∑

j qj

)a is a positive element of finite trace. Since τ is normal,

τ(a(∑

j qj)a) = τ(∑

j aqja) =∑

j τ(aqja), it follows that j : τ(aqja) 6= 0is at most countable. As τ is faithful, we deduce that the set j : aqja 6= 0is also countable. When aqja = 0, we have cqjb = 0, for every b and c inthe weak∗ closed subalgebra of N generated by a, and hence 0 = rqjr = qj ,which proves the desired statement.

It is known that the supremum of a countable family of σ-finite projectionsin N is again σ-finite (cf. [16, Exercise 5.7.45]). Thus, for any finite orcountable set F ⊂ Lp(τ), there exists a σ-finite projection p0 ∈ N suchthat φ = p0φp0, for any φ ∈ F . We summarize the results in the followinglemma.

Lemma 2.17. Suppose 1 ≤ p <∞, τ is a normal faithful semifinite trace ona von Neumann algebra N , and F is a finite or countable subset of Lp(τ).Then there exists a σ-finite projection q ∈ N , so that, for any φ ∈ F ,qφq = φ.

We shall establish now some preliminary results which are required later.Henceforth, τ denotes a faithful normal semifinite trace on a von Neumannfactor N .

Lemma 2.18. There exists a constant C = C(p), depending only on p ∈[1,∞)\2, with the following property: for any von Neumann factor N ,equipped with a normal faithful semifinite trace τ , any orthogonality-to-p-orthogonality preserving linear map T : Lp(τ)→ X, and any finite projectionr, the inequality

C−1α(T )‖φ‖p ≤ ‖T (φ)‖ ≤ Cα(T )‖φ‖p,holds for every φ ∈ Lp(rNr).

Proof. For ε > 0, find q ∈ Pτ (N) such that α(T ) + ε > ‖T (q)‖/τ(q)1p ≥

α(T |Lp(N0)). Let q0 = q ∨ r, and N0 = q0Nq0. By Theorem 2.1 for finitefactors, proved in Subsection 2.1 (see also Remark 2.3), there exists a positiveconstant C = C(p), depending only on p, such that the inequalities

C−1α(T )‖φ‖p ≤ C−1α(T |Lp(N0))‖φ‖p ≤ ‖T (φ)‖,

18 T. OIKHBERG AND A.M. PERALTA

‖T (φ)‖ ≤ Cα(T |Lp(N0))‖φ‖p < C(α(T ) + ε)‖φ‖p,hold for every φ ∈ Lp(N0). As ε > 0 is arbitrary, we are done.

We shall say that a family of projections (li, ri)i∈N ∈ Pτ (N) is adapted to(φ, p0) (here φ ∈ Lp(τ)) if:

(1) l1 ≤ l2 ≤ . . . ≤ p0 and r1 ≤ r2 ≤ . . . ≤ p0, where p0 is a σ-finiteprojection satisfying p0φp0 = φ. Furthermore, the sequences (li) and(ri) converge to p0 in the strong∗ topology of N .

(2) For every i, l⊥i φri = liφr⊥i = 0.

Below we establish the existence of adapted families.

Lemma 2.19. Suppose τ is a semifinite trace on a von Neumann algebraN , φ ∈ Lp(τ), and p0 is a σ-finite projection in N , satisfying p0φp0 = φ.Then there exists a family (li, ri)i∈N, adapted to (φ, p0). When φ∗ = φ wecan take ri = li, for every i ∈ N.

Proof. We have polar decomposition φ = u|φ|, where u is a partial isometry,whose initial and terminal projections are the orthogonal complement ofkerφ, and the closure of ranφ, respectively. For i ∈ N, let qi = 1(1/i,∞)(|φ|),and q′i = uqiu

∗. Furthermore, let q∞ = limi qi, and q′∞ = limi q′i (the

limits exist in Strong Operator Topology). Then q∞ = u∗u ≤ p0, andq′∞ = uu∗ ≤ p0. Since p0 is σ-finite, we can find increasing sequences offinite projections qi ≤ p0−q∞, and q′i ≤ p0−q′∞, such that limi qi = p0−q∞,and limi q

′i = p0− q′∞ (cf. [17, Exercise 6.9.12]). Finally, we take ri = qi + qi

and li = q′i + q′i.

Lemma 2.20. Suppose τ is a semifinite trace on a von Neumann algebraN , φ ∈ Lp(τ), and p0 is a σ-finite projection in N , satisfying p0φp0 =φ. For every increasing sequences (li)i∈N, (ri)i∈N of finite projections in Nconverging to p0 in the strong∗ topology of N , we have

limi‖φ− liφ‖p = lim

i‖φ− φri‖p = lim

i‖φ− liφri‖p = 0.

Proof. Let a be a positive element in N whose range projection, r(a), isfinite. In this case, p′0 = p0 ∨ r(a) is a σ-finite projection in N and we canfind sequences (l′i)i∈N and (r′i)i∈N of finite projections in N converging to p0

in the strong∗ topology of N such that r′i ≥ ri, l′i ≥ li and p′0ap′0 = a. Finite

linear combinations of positive elements having finite range projections aredense in Lp(τ). Thus, it suffices to prove the statements of the lemmain the case when φ is a positive element whose range projection, r(φ), isfinite and dominated by p0. That is r(φ) ∈ Pτ (N). In this case |φ− liφ|2 =|p0φ−liφ|2 = |(p0−li)φ|2 = φ(p0−li)φ is a bounded and decreasing sequencein r(φ)Nr(φ), converging to 0 in the strong∗-topology.

Let L[0,‖φ‖2] denote the set of all hermitian elements in N whose spectrum

is contained in [0, ‖φ‖2] and let f : [0, ‖φ‖2]→ C be the function defined by

f(t) = tp4 . By [29, Lemma II.4.6], the functional calculus L[0,‖φ‖2] → N , a 7→

f(a) is strongly continuous, thus |φ− liφ|p2 = f(|φ− liφ|2) converges to 0 in

AUTOMATIC CONTINUITY 19

the strong∗ topology of r(φ)Nr(φ). Noticing that τ |r(φ)Nr(φ) : r(φ)Nr(φ)→C is a finite normal trace we deduce that

‖φ− liφ‖pp = τ(|φ− liφ|p) = τ(|φ− liφ|p2 |φ− liφ|

p2 )→i 0.

Similarly, limi ‖φ−φri‖p = limi ‖(φ−φri)∗‖p = limi ‖φ∗− riφ∗‖p = 0, forevery 1 ≤ p <∞.

Finally, since φ − liφri =(φ − liφ

)+ li

(φ − φri

), the triangle inequality

shows that lim ‖φ− liφri‖p = 0.

Lemma 2.21. Suppose N is an infinite semifinite von Neumann factor,and (qi)

∞i=1 is a family of mutually orthogonal finite non-zero projections in

N , so that∑

i qi is infinite. Then we can represent N as a disjoint union ofsets A and B, so that the projections

∑i∈A qi and

∑i∈B qi are infinite.

Proof. We can find a sequence 0 = K0 < K1 < . . ., so that

∞∑j=0

K2j+1∑i=K2j+1

τ(qi) =∞∑j=0

K2j+2∑i=K2j+1+1

τ(qi) =∞.

Then set A = ∪∞j=0[K2j + 1,K2j+1] and B = ∪∞j=0[K2j+1 + 1,K2j+2].

Lemma 2.22. Suppose N is a semifinite von Neumann factor, equippedwith a normal faithful semifinite trace τ , X is a Banach space, p lies in[1,∞)\2, and T : Lp(τ) → X is a discontinuous orthogonality-to-p-orthogonality preserving linear map. Then for any θ > 0 there exists φ ∈Lp(τ)+ and a σ-finite projection r ∈ N so that rφr = φ, r⊥ is infinite,‖φ‖p ≤ 1, and ‖T (φ)‖ > θ.

Proof. Find ψ ∈ Lp(τ)+ so that ‖ψ‖p ≤ 1 and ‖T (ψ)‖ > 2θ. By Lemma2.17, there exists a σ-finite projection q0 ∈ N such that ψ = q0ψq0. Fur-thermore, by Lemma 2.19, there exits a sequence (li) of finite projectionssuch that family (li, li)i∈N is adapted to (ψ, q0). By Lemma 2.21, we canrepresent N as a disjoint union of sets A1 and A2, so that the projectionsrs =

∑i∈As(li+1−li) (s = 1, 2) are infinite. Furthermore, ψ = r1ψr1+r2ψr2,

and the two summands on the right are mutually orthogonal. Consequently,‖T (ψ)‖p =

∑2s=1 ‖T (rsψrs)‖p. Up to relabeling, we may assume that

‖T (r1ψr1)‖ > θ. Then the conclusion of the lemma holds, with r = r1,and φ = r1ψr1.

Proof of Theorem 2.1 in the semifinite case. Suppose N is a semifinite fac-tor, equipped with a normal faithful semifinite trace τ . We tackled finitefactors in Subsection 2.1. Henceforth, we assume N is not finite. As before,set Lp0(τ) = ∪e∈Pτ (N)eL

p(τ)e. By Lemma 2.18, the inequality

C−1α(T )‖φ‖p ≤ ‖T (φ)‖ ≤ Cα(T )‖φ‖p,

20 T. OIKHBERG AND A.M. PERALTA

holds for every φ ∈ Lp0(τ), where α(T ) is defined as the infimum of the

set ‖T (q)‖/τ(q)1p : q is a projection in N with τ(q) < ∞. It therefore re-

mains to prove the continuity of T and the density of Lp0(τ) in Lp(τ). Thedensity follows, from example, from Lemmas 2.19 and 2.20.

Suppose, for the sake of contradiction, that T is unbounded. By Lemma2.22, there exists a sequence of elements φn ∈ Lp(τ)+, and a sequence ofσ-finite projections pn, so that, for any n ∈ N, p⊥n is infinite, pnφnpn =φn, ‖φn‖p ≤ 1, and ‖T (φn)‖ > 4n. Moreover, for every n, find a σ-finite

projection p′n ≤ p⊥n . By [17, Exercise 5.7.45], p0 = ∨n(pn ∨ p′n) is σ-finite.By passing to p0Np0, we may assume that p0 = 1.

Find a sequence (qn) of mutually orthogonal infinite projections. By [17,Corollary 6.3.5], all infinite projections in a semifinite σ-finite factor areequivalent, hence there exist partial isometries un so that unu

∗n = qn and

u∗nun = pn. Let φ =∑∞

n=1 2−nunφnu∗n. Then, for any n,

‖T (φ)‖p =∑m 6=n

2−mp‖T (umφmu∗m)‖p + 2−np‖T (unφnu

∗n)‖p

≥ 2−np‖T (unφnu∗n)‖p.

However, by Lemma 2.13, ‖T (unφnu∗n)‖ ≥ C1(p)

− 2p ‖T (φn)‖ ≥ C1(p)

− 2p 4n

(where C1(p) is the constant given by Lemma 2.7), and therefore, ‖T (φ)‖p ≥C1(p)−22np for any n, which is impossible.

3. Disjointness preserving maps on vector-valued functionspaces

We start this section by generalizing some of the results obtained in [22,Section 3] to vector valued function spaces. Let (Ω, µ) be a σ-finite measurespace. For each t ∈ Ω, Zt is a Banach space. We say that a Banach spaceZ is a suitable function space if:

(1) The elements of Z are equivalence classes of functions z, definedalmost everywhere on Ω, with z(t) ∈ Zt almost everywhere. Twofunctions are equivalent if they coincide almost everywhere.

(2) For every z ∈ Z, and any measurable subset S ⊂ Ω, the functionzχS belongs to Z, and satisfies ‖zχS‖ ≤ ‖z‖. In other words, theprojection PS , defined by PS(z) = zχS , is contractive.

(3) limi ‖PSi(z) − z‖ = 0 whenever (Si) is an increasing sequence ofmeasurable subsets of Ω, and P∪iSi(z) = z.

We denote such a Z by∫ ⊕

Ω Zt dµ, or simply∫ ⊕

Zt dµ. For the individual

elements, we use the notation z =∫ ⊕

z(t)dµ(t).An elementary example of a suitable function space is Lp(µ, Y ), where

Y is a Banach space, and 1 ≤ p < ∞. Another example involves a non-commutative Lp space. Consider a separably acting von Neumann algebraN . Its center C can be identified with L∞(Ω, µ), where (Ω, µ) is (Ωn, νn)(1 ≤ n ≤ ∞), ([0, 1], µL), or the direct sum of the two. Here, µL is the usual

AUTOMATIC CONTINUITY 21

Lebesgue measure, and νn the counting measure on the set Ωn = 1, . . . , n(n ∈ N), or Ω∞ = N (see [17, Section 9.4]). By [17, Sections 14.1-2] (or[9, Chapter III]), there exist a field (Ht)t∈Ω of separable Hilbert spaces,and a field (Nt)t∈Ω of von Neumann algebras, such that Nt is a factor for

almost any t, and N =∫ ⊕

Nt dµ(t) (acting on H =∫ ⊕

Ht dµ(t)), where Ω

is as above. Any elements a ∈ N can be represented as∫ ⊕

a(t)dµ(t) in anessentially unique way. If N is of a certain type, then almost all the Nt’sare of the same type. If τ is a normal semifinite trace on N , we can writeτ =

∫ ⊕τt dµ(t), where τt is a normal faithful semifinite trace for almost

every t.In this setting, we can identify any element x ∈ Lp(τ) with a family x(t) ∈

Lp(τt) (for almost every t ∈ Ω), with ‖x‖pp =∫‖x(t)‖pLp(τt)

dµ. Indeed, we

know that N ∩ Lp(τ) is dense in Lp(τ). For a =∫ ⊕

a(t) ∈ N ∩ Lp(τ), weclearly have ‖a‖pp =

∫τt(|a(t)|p) dµ(t). In particular, a(t) ∈ Nt ∩ Lp(τt) for

almost every t.Any x ∈ Lp(τ) is a limit of a sequence a(n) ∈ N ∩ Lp(τ) in the norm of

Lp(τ). Write a(n) =∫ ⊕

a(n)(t)dµ(t). As limm,n

∫τt(|a(n)(t)−a(m)(t)|p) dµ =

0, we conclude that the sequence (a(n)(t)) is Cauchy in Lp(τt) for almost

every t. Therefore, for almost every t, x(t) = limn a(n)(t) exists in Lp(t)

almost everywhere, and the function t 7→ τt(|x(t)|p) is measurable. One cansee that x(t) is independent on the choice of the approximating sequence

a(n) (except possibly on a null subset of Ω).

By Fatou Lemma,∫τt(|x(t)|p) dµ(t) ≤ limn

∫τt(|a(n)(t)|p) dµ(t) = ‖x‖pp.

To prove the converse inequality, write x = uy, where u is a partial isometry,y ∈ Lp(τ)+, and u∗uyuu∗ = y. Define the functions fn(t) = mint, n (n ∈N), and let a(n) = ubn, where bn = fn(y). Then, for m ≥ n, bn = fn(bm).For almost every t, u(t) is a partial isometry, and bn(t) = fn(bm(t)). Inparticular, bn(t) ≤ bm(t). Therefore,∫

τt(|x(t)|p) dµ(t) ≥ limn

∫τt(|a(n)(t)|p) dµ(t)

= limn

∫τt(|b(n)(t)|p) dµ(t) = ‖x‖pp,

establishing the desired equality.Furthermore, if S is a measurable subset of Ω, then PS(x) represents an

element in Lp(τ), with ‖PSx‖ ≤ ‖x‖ (just consider the limit of the sequence

(PS(a(n)))). Thus, Condition (2) in the definition of a suitable space isverified. Condition (3) is dealt with in a similar manner. Thus, Lp(τ) can

be identified with the suitable function space∫ ⊕

Ω Lp(τt) dµ.

We say that x1, x2 ∈∫ ⊕

Ω Zt dµ are disjoint if ‖x1(t)‖‖x2(t)‖ = 0 almost

everywhere on Ω. A map T :∫ ⊕

Ω Zt dµ→ X is called disjointness to semi-M -orthogonality (DSMO, for short) preserving if T (x1) ⊥SM T (x2) whenever x1

22 T. OIKHBERG AND A.M. PERALTA

and x2 are disjoint. Recall that x and y are semi-M -orthogonal (x ⊥SM y)if ‖x+ y‖ ≥ max‖x‖, ‖y‖.

Theorem 3.1. (1) Suppose µ is the usual Lebesgue measure on Ω = [0, 1],

Z =∫ ⊕

Ω Zt dµ is a suitable function space, X is a Banach space, and T :Z → X is a DSMO preserving linear bijection. Then T is continuous.

(2) Suppose the set Ω is finite or countable, µ(s) ∈ (0,∞) for any s ∈ Ω,

Z =∫ ⊕

Ω Zt dµ is a suitable function space, and T : Z → X is a DSMOpreserving linear bijection, so that TPs is continuous for any s ∈ Ω. ThenT is continuous.

Start by stating the following simple lemma.

Lemma 3.2. Suppose (Ω, µ) is a measure space, X is a Banach space, Z =∫ ⊕Ω Zt dµ is a suitable function space, and T : Z → X is a DSMO preserving

linear bijection. Then, for any S ⊂ Ω of positive measure, T (PS(Z)) isclosed.

Proof. Suppose, for the sake of contradiction, that T (PS(Z)) is not closedfor some S ⊂ Ω. Then there exists a convergent sequence zi ∈ Z so that zi =∫

Ω fi(t)dµ(t), with fi(t) ∈ Zi, fi = 0 on Ω\S, but x = limi T (zi) /∈ T (PS(Z)).As T is a bijection, there exists z =

∫Ω f(t)dµ(t), so that x = T (z). For any

i, PΩ\S(zi) = 0, hence

‖T (z − zi)‖ ≥ ‖T (PΩ\S(z − zi))‖ = ‖T (PΩ\S(z))‖.

However, limi ‖T (z − zi)‖ = 0. By the bijectivity of T , PΩ\S(z) = 0. Thiscontradicts T (z) /∈ T (PS(Z)).

Below, we use the notation ES = PS(Z).

Lemma 3.3. Suppose (Ω, µ) is a measure space, X is a Banach space,

Z =∫ ⊕

Ω Zt dµ is a suitable function space, and T : Z → X is a DSMOpreserving linear map. If (Si)i∈I is a family of disjoint subsets of Ω ofpositive measure, then T |ESi is bounded for all but finitely many values of i.

Proof. Suppose, for the sake of contradiction, that there exist distinct indices(ik)k∈N, and elements zk ∈ Z, so that PSik (zk) = zk, ‖zk‖ < 2−k, and

‖T (zk)‖ > 2k. Let z =∑

k zk. Then, for any k, ‖T (z)‖ ≥ supk ‖T (zk)‖ > 2k,which is impossible.

Lemma 3.4. Suppose (Ω, µ) is a measure space, X is a Banach space,

Z =∫ ⊕

Ω Zt dµ is a suitable function space, and T : Z → X is a DSMOpreserving linear map. Suppose, furthermore, that (Si)i∈I is a family ofdisjoint subsets of Ω, of positive measure, so that T |ESi is bounded for anyi. Then there exists a constant C > 0 so that, for any finite F ⊂ I, we have

‖T |E∪i∈F Si‖ ≤ C.

AUTOMATIC CONTINUITY 23

Proof. Suppose otherwise. Then there exists a family (Fn) of disjoint finitesubsets of I, so that, for any n,

‖T |P∪i∈FnSi (Z)‖ > 4n.

Thus, there exist zn ∈ P∪i∈FnSi(Z), so that ‖zn‖ < 2−n, while ‖T (zn)‖ > 2n.We complete the proof by noting that ‖T (

∑n zn)‖ > ‖T (zn)‖ for every

n.

Lemma 3.5. Suppose that, in the notation of Lemma 3.4, µ is σ-finite, andT is bijective. Then T is bounded on E∪i∈ISi.

Proof. Note first that the collection (Si) is at most countable. If I is finite,an application of Lemma 3.4 completes the proof. Otherwise, we can assumeI = N. By Lemma 3.4, there exists C > 0 so that ‖TE∪n

i=1Si‖ ≤ C for any

n ∈ N. Denote S = ∪i∈NSi, and show that ‖TES‖ ≤ C. Pick z ∈ ES . For thesake of brevity, let Qn = P(Ω\S)∪(∪ni=1Sn). Let zn = Qn(z). By the definition

of a suitable function space, limn zn = z. However, ‖T (zn) − T (zm)‖ ≤C‖zn − zm‖, hence the sequence (T (zn)) converges to some x ∈ X. By thebijectivity of T , there exists y ∈ Z so that T (y) = x. We have to show thaty = z. For this, it suffices to prove that, for every n, Qn(y) = zn. For anym > n, QmQn = Qn. Then

y−zm = Qn(y−zm)+(1−Qn

)(y−zm) =

(Qn(y)−zn

)+(1−Qn

)(y−zm),

hence ‖T (y)− T (zm)‖ ≥ ‖T (Qn(y)− zn)‖. Passing to the limit as m→∞,we see that ‖T (Qn(y) − zn)‖ = 0. The bijectivity of T yields the desiredconclusion.

Proof of Theorem 3.1. (1) For convenience of notation, we denote by Σ+ theset of all measurable sets S ⊂ [0, 1] with µ(S) > 0. Denote by Σ′ the setof equivalence classes of sets from S ∈ Σ+ (modulo sets of measure 0). LetS be the set of all equivalence classes [S] ∈ Σ′ for which T is bounded onES (clearly, this definition does not depend on the choice of a representativefrom an equivalence class). We shall show that [Ω] ∈ S.

First show that any S ∈ Σ+ contains a subset S′ of positive measure,belonging to S. Indeed, write S as a disjoint union of sets Sk ∈ Σ+. ByLemma 3.3, T is bounded on ESk , for all but finitely many values of k.

Next observe that, if the sets Sk ∈ Σ+ (k ∈ N) are such that T is boundedon ESk for every k, then T is bounded on E∪kSk . Indeed, by passing fromSk to Sk\ ∪j<k Sj if necessary, we can assume that the sets Sk are disjoint.Then apply Lemma 3.5.

Define a partial order on S by writing [S1] ≺ [S2] if µ(S2\S1) ≥ 0 andµ(S1\S2) = 0. Any chain in S has an upper bound. Indeed, any such chaincan have at most countably many distinct elements, due to the σ-finitenessof µ. We have observed above that [∪kSk] ∈ S whenever [Sk] ∈ S for everyk (cf. Lemma 3.5). Thus, by Zorn’s Lemma, S has at least one maximalelement. But [Ω] is the only possible maximal element.

(2) Follows directly from Lemma 3.5.

24 T. OIKHBERG AND A.M. PERALTA

We can now establish the automatic continuity of orthogonality-to-p-orthogonality preserving linear bijections from a non-commutative Lp(τ)space associated with a separably acting von Neumann algebra admitting asemifinite normal faithful trace.

Theorem 3.6. Suppose N is a separably acting von Neumann algebra,equipped with a normal faithful semifinite trace τ , X is a Banach space,p ∈ [1,∞)\2, and T : Lp(τ) → X is an orthogonality-to-p-orthogonalitypreserving linear bijection. Then T is continuous.

Proof. Represent N as a direct sum:

N = (⊕k=1,2,...,∞NIk)⊕NII1 ⊕NII∞ ,

with the algebras NIk , NII1 , and NII∞ having type Ik, II1, and II∞, re-spectively. We have to show the continuity of T on each of these sum-mands. The summand of type α can be represented as a direct integralNα =

∫ ⊕Nα,t dµα(t), where µα is an appropriate measure on a set Ωα,

arising from the center of Nα (cf. Sections 14.1 and 14.2 in [17]). Write

Ωα = Ω(d)α ∪ Ω

(c)α , where Ω

(d)α is either finite or countable, and Ω

(c)α is ei-

ther empty, or [0, 1] (see [17, Section 9.4]). The restrictions of µα on these

subsets is denoted by µ(d)α and µ

(c)α , respectively. The direct integral de-

composition (explained in the beginning of Section 3) allows us to write

N(i)α =

∫Ω

(i)αNα(t)dµα (i ∈ c, d). Denote the restriction of τ to N

(i)α by

τ(i)α , we write Lp(τ) as an orthogonal sum of terms Lp(τ

(i)α ). As noted in the

beginning of this section, Lp(τ) is a suitable function space.

By Theorem 2.1 (combined with Lemma 3.2), for any α and any s ∈ Ω(d)α ,

TPs is bounded. Thus, by Theorem 3.1(2), T is bounded on(∑

α Lp(τ

(d)α ))p.

By Theorem 3.1(1), T is bounded on(∑

α Lp(τ

(c)α ))p.

References

[1] J. Araujo, K. Jarosz, Biseparating maps between operator algebras, J. Math. Anal.Appl. 282, no. 1, 48-55 (2003).

[2] W. Arendt, Spectral properties of Lamperti operators, Indiana Univ. Math. J. 32,no. 2, 199-215 (1983).

[3] R. Bhatia, Matrix Analysis, Springer-Verlag, Berlin, 1997.[4] A. Bikchentaev, O. Tikhonov, Characterization of the trace by monotonicity inequal-

ities, Linear Algebra Appl. 422, no. 1, 274-278 (2001).[5] A. Blanco, A. Turnsek, On maps that preserve orthogonality in normed spaces, Proc.

Roy. Soc. Edinburgh Sect. A 136, no. 4, 709-716 (2006).[6] M. Burgos, F.J. Fernandez-Polo, J.J. Garces, J. Martınez Moreno, A.M. Peralta,

Orthogonality preservers in C*-algebras, JB*-algebras and JB*-triples, J. Math. Anal.Appl. 348, 220-233 (2008).

[7] M. Burgos, F.J. Fernandez-Polo, J.J. Garces, A.M. Peralta, Orthogonality preserversrevisited, Asian-Eur. J. Math. 2 (3), 387-405 (2009).

[8] M. Burgos, J. Garces, A.M. Peralta, Automatic continuity of biorthogonality pre-servers between compact C*-algebras and von Neumann algebras, J. Math. Anal.Appl. 376, 221-230 (2011).

AUTOMATIC CONTINUITY 25

[9] J. Dixmier, von Neumann algebras, North Holland, New York (1981).[10] T. Fack, H. Kosaki, Generalized s-numbers of τ -measurable operators, Pacific J.

Math. 123, 269-300 (1986).[11] J. Gleason, Measures on the closed subspaces of a Hilbert space, J. Math. Mech. 6,

885-893 (1957).[12] S. Goldstein, A. Paszkiewicz, Linear combinations of projections in von Neumann

algebras, Proc. Amer. Math. Soc. 116, 175-183 (1992).[13] S. Hernandez, E. Beckenstein, L. Narici, Banach-Stone theorems and separating

maps, Manuscripta Math. 86, no. 4, 409-416 (1995).[14] K. Jarosz, Automatic continuity of separating linear isomorphisms, Canad. Math.

Bull. 33, 139-144 (1990).[15] J.-S. Jeang, N.-Ch. Wong, Weighted composition operators of C0(X)’s, J. Math.

Anal. Appl. 201, 981-993 (1996).[16] R.V. Kadison, J. Ringrose, Fundamentals of the Theory of Operator Algebras I, Aca-

demic Press, 1983.[17] R.V. Kadison, J. Ringrose, Fundamentals of the Theory of Operator Algebras II,

Academic Press, 1985.[18] Ch.-W. Leung, N.-Ch. Wong, Zero product preserving linear maps of CCR C∗-

algebras with Hausdorff spectrum, J. Math. Anal. Appl. 361, no. 1, 187-194 (2010).[19] Ch.-W. Leung, Ch.-W. Tsai, N.-Ch. Wong, Separating linear maps of continuous

fields of Banach spaces, Asian-Eur. J. Math. 2, no. 3, 445-452 (2009).[20] E. Nelson, Notes on non-commutative integration, J. Funct. Anal. 15, 103-116 (1974).[21] T. Oikhberg, A.M. Peralta, M. Ramırez, Automatic continuity of M-norms on C*-

algebras, J. Math. Anal. Appl. 381, 799-811 (2011).[22] T. Oikhberg, A.M. Peralta, D. Puglisi, Automatic continuity of orthogonality or dis-

jointness preserving bijections, to appear in Rev. Mat. Complut.. DOI 10.1007/s13163-011-0089-0.

[23] B. de Pagter, Non-commutative Banach function spaces, Positivity, Trends Math.,Birkhauser, Basel, 2007, pp. 197–227.

[24] A. Paszkiewicz, Measures on projections in W ∗-factors, J. Funct. Anal. 62, 87-117(1985).

[25] G. Pisier, Q. Xu, Non-commutative Lp-spaces, in Handbook of the geometry of Banachspaces, Vol. 2, 1459-1517, North-Holland, Amsterdam, 2003.

[26] Y. Rayndaud, Q. Xu, On subspaces of non-commutative Lp spaces, J. Funct. Anal.203, 149-196 (2003).

[27] K.S. Saito, Noncommutative Lp-spaces with 0 < p < 1, Math. Proc. CambridgePhilos. Soc. 89, 405-411 (1981).

[28] A. Sherstnev, A. Stolyarov, O. Tikhonov, Characterization of normal traces on vonNeumann algebras by inequalities for the modulus, Math. Notes 72, 411-416 (2002).

[29] M. Takesaki, Theory of operator algebras I, Springer, New York, 2003.[30] M. Takesaki, Theory of operator algebras II, Springer, New York, 2003.[31] O. Tikhonov, Subadditivity inequalities in von Neumann algebras and characteriza-

tion of tracial functionals, Positivity 9, 259-264 (2005).[32] Ch.-W. Tsai, The orthogonality structure determines a C∗-algebra with continuous

trace, Oper. Matrices 5, No. 3, 529-540 (2011).[33] M. Wolff, Disjointness preserving operators in C∗-algebras, Arch. Math. 62, 248-253

(1994).[34] S. Yamagami, Notes on operator categories, J. Math. Soc. Japan 59, 541-555 (2007).[35] E.J. Yeadon, Non-commutative Lp-spaces, Math. Proc. Cambridge Philos. Soc. 77,

91-102 (1975).

26 T. OIKHBERG AND A.M. PERALTA

Dept. of Mathematics, University of California - Irvine, Irvine CA 92697,and, Dept. of Mathematics, University of Illinois at Urbana-Champaign, Ur-bana IL 61801

E-mail address: toikhber@math.uci.edu

Departamento de Analisis Matematico, Universidad de Granada,, Facultadde Ciencias 18071, Granada, Spain

E-mail address: aperalta@ugr.es