In this note we show that Kadison's similarity problem for C*-algebras is
equivalent to a problem in perturbation theory: must close C*-algebras have
close commutants?Comment: 6 Pages, minor typos fixed. C. R. Acad. Sci. Canada, to appea

Cameron, Jan

Christensen, Erik

Sinclair, Allan M.

Smith, Roger R.

White, Stuart

Wiggins, Alan D.

English

arXiv

In this note we show that Kadison's similarity problem for C*-algebras is equivalent to a problem in perturbation theory: must close C*-algebras have close commutants

Cameron, J.

Christensen, E.

Sinclair, A.M.

Smith, R.R.

White, S.

Wiggins, A.D.

Enlighten

          Cameron, J., Christensen, E., Sinclair, A.M., Smith, R.R., White, S., and Wiggins, A.D. (2013) A remark on the similarity and perturbation problems. Comptes Rendus Mathematiques de l'Académie des Sciences. ISSN 0706-1994     Copyright © 2013 The authors    A copy can be downloaded for personal non-commercial research or study, without prior permission or charge   The content must not be changed in any way or reproduced in any format or medium without the formal permission of the copyright holder(s)   When referring to this work, full bibliographic details must be given    http://eprints.gla.ac.uk/73281/      Deposited on: 4 February 2013                  Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk A REMARK ON THE SIMILARITY AND PERTURBATION PROBLEMSJAN CAMERON, ERIK CHRISTENSEN, ALLAN M. SINCLAIR, ROGER R. SMITH,STUART WHITE, AND ALAN D. WIGGINSAbstract. In this note we show that Kadison’s similarity problem for C∗-algebras is equiv-alent to a problem in perturbation theory: must close C∗-algebras have close commutants?Let A be a C∗-algebra. In 1955 Kadison asked whether every bounded homomorphismσ : A → B(H) is similar to a ∗-homomorphism [12], i.e. does there exist an invertibleoperator S ∈ B(H) such that S−1σ(·)S is a ∗-homomorphism? This problem, which remainsopen, has positive answers in the following cases: for amenable algebras [2], for properlyinfinite von Neumann algebras and traceless C∗-algebras [10], in the presence of a cyclicvector [10], and for II1 factors with property Γ [8]. It is equivalent to a number of importantproblems and properties:• is every bounded homomorphism σ : A→ B(H) completely bounded? ([10])• the derivation problem [15] - given a faithful non-degenerate representation ι : A →B(H), is every derivation of ι(A) into B(H) spatial?• the distance property [5, 7, 15] - does there exist a constant K > 0 such that givenany faithful non-degenerate representation ι : A→ B(H)(1)12‖ad(T )|ι(A)‖cb = d(T, ι(A)′) ≤ K‖ad(T )|ι(A)‖holds for all T ∈ B(H)?1 When this holds, A is said to have property DK . Here, andthroughout the paper, ad(T ) denotes the spatial derivation X 7→ [T,X] = TX −XTon B(H).• do all C∗-algebras have finite length?2 ([17])• are all von Neumann algebras hyperreflexive? (see [18, Theorem 10.3]).• Diximer’s invariant operator range problem from 1950 (see [18, Problem 10.4]).In this note we add to the list of equivalent formulations by showing that the similarityproblem is equivalent to an open question in the uniform perturbation theory of operatoralgebras.The Kadison-Kastler metric on all C∗-subalgebras of B(H) is given by restricting theHausdorff metric on subsets of B(H) to the unit balls. Specifically, for pairs of C∗-algebrasDate: June 22, 2012.JC’s research is partially supported by an AMS-Simons research travel grant.RS is partially supported by NSF grant DMS-1101403.SW is partially supported by EPSRC grant EP/I019227/1.1The equality in (1) is Arveson’s distance formula from [1] and is always valid.2Pisier’s characterisation of the similarity problem using his notion of length provides an intrinsic formu-lation of the similarity property in terms of internal behaviour.12 CAMERON, CHRISTENSEN, SINCLAIR, SMITH, WHITE, AND WIGGINSA,B ⊆ B(H),d(A,B) = max supx∈A‖x‖≤1 infy∈B‖y‖≤1 ‖x− y‖, supy∈B‖y‖≤1 infx∈A‖y‖≤1 ‖y − x‖ .This notion was introduced in [13] which conjectures that sufficiently close von Neumannalgebras must arise from a small unitary perturbation. Precisely, for all ε > 0, there shouldexist δ > 0 such that, given von Neumann algebras M,N ⊆ B(H) with d(M,N) < δ, thereexists a unitary U on H with ‖U − IH‖ < ε satisfying UMU∗ = N.3 This conjecture, if true,immediately implies that the operation of taking commutants of C∗-algebras is continuouswith respect to the Kadison-Kastler metric.4 Establishing that certain close operator algebrashave close commutants, and a related property of near inclusions (see Remark 2 below) playa key role in our recent work on the Kadison-Kastler problem for crossed product factors [3],in EC’s solution to the near inclusion problem for injective von Neumann algebras [6] and inwork on transferring K-theoretic invariants between close algebras [14, 9, 16] (see Remark 3below). Here we show that the continuity of the operation of taking commutants is equivalentto the similarity problem. In order to phrase this result for individual C∗-algebras we makethe following definition.Definition. Let A be a C∗-algebra. Say that commutants are continuous at A if, for ε > 0,there exists δ > 0 such that given a faithful non-degenerate ∗-representation ι : A → B(H)and a C∗-algebra B ⊆ B(H) with d(ι(A),B) < δ, we have d(ι(A)′,B′) < ε.Theorem. The following statements are equivalent for a C∗-algebra A:(I) A satisfies Kadison’s similarity property;(II) commutants are continuous at A.Further, the similarity problem has a positive answer for all C∗-algebras if and only if theoperation of taking commutants on B(H) is continuous in the case when H is an infinitedimensional separable Hilbert space.Proof of the TheoremIf the similarity problem has a positive answer for all C∗-algebras, then there exists aconstant K > 0 such that all C∗-algebras have property DK (by [15] and [7]). It is then easyto see that(2) d(A′,B′) ≤ 4Kd(A,B)for all C∗-algebras A,B ⊂ B(H) (this dates back to [6], and can be found explicitly bycombining Proposition 2.3 (iii) and Proposition 2.5 of [9]). The local version (I) =⇒ (II)of this implication is much more involved as we do not have the information that B alsohas property DK required in the previous argument. Nevertheless, this implication wasestablished in [9, Theorem 4.2] as a step towards showing that ifA has the similarity property,then any C∗-algebra sufficiently close to A also has the similarity property.3Due to the results of [4, 11], the analogous conjecture for C∗-algebras is that sufficiently close separableC∗-algebras acting on a separable Hilbert space should be spatially isomorphic; one cannot demand controlon the distance from an implementing unitary to the identity in this context.4Via a Kaplansky density argument, it suffices to consider the continuity of taking commutants on theset of von Neumann subalgebras of B(H).A REMARK ON THE SIMILARITY AND PERTURBATION PROBLEMS 3We now turn to the reverse implication (II) =⇒ (I). For X,R ∈ B(H), we use the identityRmX − XRm = Rm−1(RX − XR) + (Rm−1X − XRm−1)R to establish by induction theestimates(3) ‖[Rm, X]‖ ≤ m‖R‖m−1‖[R,X]‖, m ∈ N.Suppose that commutants are continuous at A, but that there exists no constant K > 0such that A has property DK . Then we may find a sequence (ιn)n of faithful, non-degeneraterepresentations ι : A→ B(Hn) and operators Tn ∈ B(Hn) withd(Tn, ιn(A)′) > n‖ad(Tn)|ιn(A)′‖, n ≥ 1.For each n, choose S˜n ∈ ιn(A)′ satisfying ‖S˜n − Tn‖ = d(Tn, ιn(A)′). Write Sn = Tn − S˜n sothat(4) ‖Sn‖ > n‖ad(Sn)|ιn(A)′‖, n ≥ 1.Scaling in this inequality allows us to assume that ‖Sn‖ = 1 while retaining the valid-ity of (4). Since d(Sn, ιn(A)′) = 1, either d(<(Sn), ιn(A)′) ≥ 1/2 or d(=(Sn), ιn(A)′) ≥1/2 for each n, while the equation ‖ad(Sn)|ιn(A)‖ = ‖ad(S∗n)|ιn(A)‖ gives the limiting be-haviour ‖ad(<(Sn))|ιn(A)‖, ‖ad(=(Sn))|ιn(A)‖ → 0 as n → ∞. Thus we obtain a sequenceof self-adjoint contractions Rn ∈ B(Hn) such that d(Rn, ιn(A)′) ≥ 1/2 for n ≥ 1 andlimn→∞ ‖ad(Rn)|ιn(A)‖ = 0. By approximating f(t) =√1− t2 uniformly by polynomialson [−1, 1] and using the estimate (3), we see that the unitary operatorsU±n = Rn ± i√1−R2nsatisfy limn→∞ ‖ad(U±n )|ιn(A)‖ = 0. SinceRn = (U+n +U−n )/2 for each n, either d(U+n , ιn(A)′) ≥1/2 or d(U−n , ιn(A)′) ≥ 1/2. Appropriate choices of signs give unitaries Un ∈ B(Hn) with(5) d(Un, ιn(A)′) ≥ 1/2, n ≥ 1,while limn→∞ ‖ad(Un)|ιn(A)‖ = 0.Now consider the faithful non-degenerate representations ιn ⊕ ιn : A → M2(B(Hn)) sothat(ιn ⊕ ιn)(A) ={(ιn(a) 00 ιn(a)): a ∈ A}.Define C∗-algebras Bn byBn ={(ιn(a) 00 Unιn(a)U∗n): a ∈ A}⊆M2(B(Hn)), n ≥ 1,so that d((ιn ⊕ ιn)(A),Bn) ≤ ‖ad(Un)|ιn(A)‖ → 0. Clearly Wn =(0 IHn0 0)∈ (ιn ⊕ ιn)(A)′so, by hypothesis, d(Wn,B′n) → 0. Choose W˜n ∈ B′n with ‖Wn − W˜n‖ = d(Wn,B′n) and letXn be the (1, 2)-entry of W˜n. Then ‖Xn − IHn‖ → 0 andιn(a)Xn = XnUnιn(a)U∗n, a ∈ A.Thus XnUn ∈ ιn(A)′ so that d(Un, ιn(A)′)→ 0, contradicting (5). This completes the proofof (II) =⇒ (I).Finally, note that Kadison’s similarity problem is a equivalent to the existence of a constantK > 0 such that12‖ad(T )|A‖cb ≤ K‖ad(T )|A‖, T ∈ B(H),4 CAMERON, CHRISTENSEN, SINCLAIR, SMITH, WHITE, AND WIGGINSwhenever H is a separable Hilbert space and A ⊆ B(H) is a C∗-algebra. This is establishedby a standard argument (cf the last two paragraphs of the proof of [3, Proposition 2.1.5])which shows that this last condition implies the same condition without the separabilityassumption. Consequently, the proof of (II) =⇒ (I) shows that if the operation of takingcommutants is continuous on B(H) for H separable and infinite dimensional, then Kadison’ssimilarity property holds.Remarks1. If the operation of taking commutants is globally continuous, then there is a universalconstant K > 0 such that all C∗-algebras have property DK and hence (2) holds. In partic-ular taking commutants is Lipschitz. In the local situation, the proof of (I) =⇒ (II) given in[9, Theorem 4.2] shows that if A ⊆ B(H) is a C∗-algebra which has finite similarity length` and length constant K, then there exists a constant C ′`,K such thatd(A′,B′) ≤ C ′`,Kd(A,B)for all C∗-algebras B on H with d(A,B) sufficiently small. Since the metric d has diameter1, it follows that there is a constant C`,K such thatd(A′,B′) ≤ C`,Kd(A,B)for all C∗-algebras B on H. This says that if A has the similarity property, then takingcommutants is locally a Lipschitz operation near A.2. The similarity property is also characterised by the property that near inclusions ofalgebras give rise to near inclusions of their commutants. For C∗-algebras A,B ⊆ B(H)and γ ≥ 0, write A ⊆γ B if, given x ∈ A, there exists y ∈ B with ‖x − y‖ ≤ γ‖x‖. It isnatural to ask whether, given ε > 0, there exists δ > 0 such that the near inclusion A ⊆δ Bimplies that B′ ⊆ε A′.5 Indeed the connection between the similarity problem and questionsof close commutants originates in [6] where a positive answer is given when A has propertyDK . This question is also equivalent to the similarity property, and the local version at A isequivalent to the similarity property for A. The global statement follows immediately, as theability to take commutants of all near inclusions in this way implies that taking commutantsis continuous. For the local statement, note that the proof of the theorem only uses thecontinuity of commutants for pairs (A, UAU∗), and this follows from being able to takecommutants of near inclusions of A.3. It is natural to consider a completely bounded version of the Kadison-Kastler metric:dcb(A,B) = supn d(A ⊗Mn, B ⊗Mn).6 When dcb(A,B) is sufficiently small, one can showthat A and B have the same K-theoretic invariants used in the classification programme,[14, 9]. Moreover in this case A and B also have isomorphic Cuntz semigroups [16]. Thus it isof interest to learn when d(·, ·) and dcb(·, ·) are equivalent metrics: this too is a reformulationof the similarity problem.7 That the similarity problem implies the equivalence of these5One can also formulate this question locally at A by asking that this hold under all faithful non-degeneraterepresentations of A.6Similarly define A ⊂cb,γ B if and only if A⊗Mn ⊂γ B⊗Mn for all n ≥ 1.7A local version of this statement also holds: A has the similarity property if and only if dcb(A, ·) isequivalent to d(A, ·).A REMARK ON THE SIMILARITY AND PERTURBATION PROBLEMS 5metrics is noted at the end of [9, Section 4]. The converse follows as Arveson’s distanceformula shows that A ⊂cb,γ B =⇒ B′ ⊂cb,γ A′ (see [3, Proposition 2.2.3]) and sod(A′,B′) ≤ dcb(A′,B′) ≤ 2dcb(A,B).Acknowledgements. When SW presented the work of [3] in Vanderbilt University in Jan-uary 2012, Dietmar Bisch asked whether it was necessary to use the similarity property inorder to take commutants of close C∗-algebras. This directly sparked the present paper andwe are grateful to Dietmar for bringing this question to our attention.References[1] William B. Arveson, Interpolation problems in nest algebras, J. Funct. Anal. 20 (1975), 208–233.[2] John W. Bunce, The similarity problem for representations of C∗-algebras, Proc. Amer. Math. Soc. 81(1981), no. 3, 409–414.[3] Jan Cameron, Erik Christensen, Allan M. Sinclair, Roger R. Smith, Stuart White, and Alan D. Wiggins,Kadison-Kastler stability of II1-factors, Preprint, 2012.[4] Man-Duen Choi and Erik Christensen, Completely order isomorphic and close C∗-algebras need not be∗-isomorphic, Bull. London Math. Soc. 15 (1983), no. 6, 604–610.[5] Erik Christensen, Extension of derivations, J. Funct. Anal. 27 (1978), no. 2, 234–247.[6] , Near inclusions of C∗-algebras, Acta Math. 144 (1980), no. 3-4, 249–265.[7] , Extensions of derivations. II, Math. Scand. 50 (1982), no. 1, 111–122.[8] , Similarities of II1 factors with property Γ, J. Operator Theory 15 (1986), no. 2, 281–288.[9] Erik Christensen, Allan M. Sinclair, Roger R. Smith, and Stuart White, Perturbations of C∗-algebraicinvariants, Geom. Funct. Anal. 20 (2010), no. 2, 368–397.[10] Uffe Haagerup, Solution of the similarity problem for cyclic representations of C∗-algebras, Ann. ofMath. (2) 118 (1983), no. 2, 215–240.[11] Barry E. Johnson, A counterexample in the perturbation theory of C∗-algebras, Canad. Math. Bull. 25(1982), no. 3, 311–316.[12] Richard V. Kadison, On the othogonalization of operator representations, Amer. J. Math. 77 (1955),600–620.[13] Richard V. Kadison and Daniel Kastler, Perturbations of von Neumann algebras. I. Stability of type,Amer. J. Math. 94 (1972), 38–54.[14] Mahmood Khoshkam, Perturbations of C∗-algebras and K-theory, J. Operator Theory 12 (1984), no. 1,89–99.[15] Eberhard Kirchberg, The derivation problem and the similarity problem are equivalent, J. OperatorTheory 36 (1996), no. 1, 59–62.[16] Fransesc Perera, Andrew S. Toms, Stuart White, and Wilhelm Winter, The Cuntz semigroup of closeC∗-algebras, Manuscript in preparation, 2012.[17] Gilles Pisier, The similarity degree of an operator algebra, Algebra i Analiz 10 (1998), no. 1, 132–186.[18] , Similarity problems and completely bounded maps, expanded ed., Lecture Notes in Mathematics,vol. 1618, Springer-Verlag, Berlin, 2001, Includes the solution to “The Halmos problem”.6 CAMERON, CHRISTENSEN, SINCLAIR, SMITH, WHITE, AND WIGGINSJan Cameron, Department of Mathematics, Vassar College, Poughkeepsie, New York, U.S.A.E-mail address: jacameron@vassar.eduErik Christensen, Institute for Mathematiske Fag, University of Copenhagen, Copenhagen,Denmark.E-mail address: echris@math.ku.dkAllan M. Sinclair, School of Mathematics, University of Edinburgh, JCMB, King’s Build-ings, Mayfield Road, Edinburgh, EH9 3JZ, Scotland.E-mail address: a.sinclair@ed.ac.ukRoger R. Smith, Department of Mathematics, Texas A&M University, College Station,Texas, 77843, U.S.A.E-mail address: rsmith@math.tamu.eduStuart White, School of Mathematics and Statistics, University of Glasgow, UniversityGardens, Glasgow Q12 8QW, Scotland.E-mail address: stuart.white@glasgow.ac.ukAlan D. Wiggins, Department of Mathematics and Statistics. University of Michigan-Dearborn, 4901 Evergreen Road, Dearborn, Michigan, 48126, USA.E-mail address: adwiggin@umd.umich.edu

A remark on the similarity and perturbation problems

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