Turbulence, orbit equivalence, and the classification of nuclear <i>C<sup>*</sup>
                  </i>-algebras

Farah, Ilijas; Toms, Andrew S.; Törnquist, Asger

doi:10.1515/crelle-2012-0053

Cited by 25 publications

(56 citation statements)

References 24 publications

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“…We record a consequence on the classification problem for strongly selfabsorbing algebras (see [17] for the definitions). While the isomorphism relation for separable C * -algebras (and even separable AI algebras) is not classifiable by countable structures ( [18]), our result shows that the isomorphism relation for strongly self-absorbing algebras is much simpler. Since the computation of a theory of a C * -algebra is given by a Borel function ( [17]) the following is an immediate consequence of Theorem 2.16.…”

Section: 4mentioning

confidence: 73%

Relative commutants of strongly self-absorbing $$\mathrm {C}^*$$ C ∗ -algebras

Farah

Hart

Rørdam

et al. 2016

Sel. Math. New Ser.

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Abstract. The relative commutant A ∩A U of a strongly self-absorbing algebra A is indistinguishable from its ultrapower A U . This applies both to the case when A is the hyperfinite II1 factor and to the case when it is a strongly self-absorbing C * -algebra. In the latter case we prove analogous results for ∞(A)/c0(A) and reduced powers corresponding to other filters on N. Examples of algebras with approximately inner flip and approximately inner half-flip are provided, showing the optimality of our results. We also prove that strongly self-absorbing algebras are smoothly classifiable, unlike the algebras with approximately inner halfflip.Most uses of ultrapowers in operator algebras and elsewhere rely on two of their model-theoretic properties: countable saturation and Loś' theorem, stating that the canonical embedding of a structure in its ultrapower is elementary.1 These conditions -saturation and elementary equivalence -characterize the ultrapower under the assumption of the continuum hypothesis: two elementary extensions of density character ℵ 1 of a separable structure A which are saturated are necessarily isomorphic over A. This is the uniqueness theorem for saturated models; see §1.1.2 and in particular Theorem 1.2. For an arbitrary metric structure A, we can identify A with its diagonal image in an ultrapower A U and provided that A has a multiplicative structure consider the central sequence algebra (also known as the relative commutant of A in its ultrapower)When A is a C * -algebra, the relevant notion of ultraproduct is the C * -algebra ultraproduct and when A is a II 1 factor and the notion of ultraproduct is the tracial ultraproduct of von Neumann algebras. When A is a unital C * -algebra, the central sequence algebra is denoted We are indebted to Leonel Robert for proving (2) of Theorem 1 and giving a large number of very useful remarks.1 All ultrafilters are assumed to be nonprincipal ultrafilters on N and A U denotes the ultrapower of A associated with U.

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Section: 4mentioning

confidence: 73%

Relative commutants of strongly self-absorbing $$\mathrm {C}^*$$ C ∗ -algebras

Farah

Hart

Rørdam

et al. 2016

Sel. Math. New Ser.

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“…This framework captures the vast majority of concrete classification results in mathematics. (In [11] and [12] the computation of most of the invariants in the theory of C*-algebras is shown to be Borel. )…”

Section: Introductionmentioning

confidence: 99%

“…Many nonclassifiability results were established directly or indirectly using this criterion. For instance Hjorth showed in [17] (Section 4.3) that the orbit equivalence relation of a turbulent Polish group action is Borel reducible to the relation of homeomorphism of compact spaces, which in turn is reducible to the relation of isomorphism of separable simple nuclear unital C*-algebras by a result of Farah-Toms-Törnquist (Corollary 5.2 of [11]). As a consequence these equivalence relations are not classifiable by countable structures.…”

Section: Introductionmentioning

confidence: 99%

Unitary equivalence of automorphisms of separable C*-algebras

Lupini

2014

Advances in Mathematics

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Abstract. We prove that the automorphisms of any separable C*-algebra that does not have continuous trace are not classifiable by countable structures up to unitary equivalence. This implies a dichotomy for the Borel complexity of the relation of unitary equivalence of automorphisms of a separable unital C*-algebra: Such relation is either smooth or not even classifiable by countable structures. IntroductionIf A is a separable C*-algebra, the group Aut(A) of automorphisms of A is a Polish group with respect to the topology of pointwise norm convergence. An automorphism of A is called (multiplier) inner if it is induced by the action by conjugation of a unitary element of the multiplier algebra M (A) of A. Inner automorphisms form a Borel normal subgroup Inn(A) of the group of automorphisms of A. The relation of unitary equivalence of automorphisms of A is the coset equivalence relation on Aut(A) determined by Inn(A). (The reader can find more background on C*-algebras in Section 2.) The main result presented here asserts that if A does not have continuous trace, then it is not possible to effectively classify the automorphisms of A up to unitary equivalence using countable structures as invariants; in particular this rules out classification by K-theoretic invariants. In the course of the proof of the main result we will show that the existence of an outer derivation on a C*-algebra A is equivalent to a seemingly stronger statement, that we will refer to as Property AEP (see Definition 4.4), implying in particular the existence of an outer derivable automorphism of A.The notion of effective classification can be made precise by means of Borel reductions in the framework of descriptive set theory (the monographs [20] and [15] are standard references for this subject). If E and E ′ are equivalence relations on standard Borel spaces X and X ′ respectively, then a Borel reduction from E to F is a Borel function f : X → X ′ such that for every x,y ∈ X, xEy if and only if f (x) E ′ f (y). The Borel function f witnesses an effective classification of the objects of X up to E, with E ′ -equivalence classes of objects of X ′ as invariants. This framework captures the vast majority of concrete classification results in mathematics. (In [11] and [12] the computation of most of the invariants in the theory of C*-algebras is shown to be Borel.)2010 Mathematics Subject Classification. 03E15, 46L40, 46L57.

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“…Let K Choq denote the convex subsets of Q that are Choquet simplexes. In Section 4 of [FTT14], it is shown that K Choq is a Borel subset of K(Q), and may be taken to be the standard Borel space of metrizable Choquet simplexes. In [Sab13], Sabok shows that E grp ∼ B ≈ a , the relation of affine homeomorphism on K Choq .…”

Section: Homeomorphism Of Compact Metric Spacesmentioning

confidence: 99%

“…Later, J. Melleray in [Mel07] demonstrated that the classification of separable Banach spaces by linear isometries is also complete in this class. More recently, in [FTT14] and [EFP + 13], the isomorphism relation on separable C*-algebras was seen to be a member of this family as well, and it too was proven to be complete in [Sab13].…”

Section: Introductionmentioning

confidence: 99%

The complexity of the homeomorphism relation between compact metric spaces

Zielinski

2016

Advances in Mathematics

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Abstract. We determine the exact complexity of classifying compact metric spaces up to homeomorphism. More precisely, the homeomorphism relation on compact metric spaces is Borel bi-reducible with the complete orbit equivalence relation of Polish group actions. Consequently, the same holds for the isomorphism relation between separable commutative C*-algebras and the isometry relation between C(K)-spaces.

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Turbulence, orbit equivalence, and the classification of nuclear C^* -algebras

Cited by 25 publications

References 24 publications

Relative commutants of strongly self-absorbing $$\mathrm {C}^*$$ C ∗ -algebras

Relative commutants of strongly self-absorbing $$\mathrm {C}^*$$ C ∗ -algebras

Unitary equivalence of automorphisms of separable C*-algebras

The complexity of the homeomorphism relation between compact metric spaces

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Turbulence, orbit equivalence, and the classification of nuclear C* -algebras

Cited by 25 publications

References 24 publications

Relative commutants of strongly self-absorbing $$\mathrm {C}^*$$ C ∗ -algebras

Relative commutants of strongly self-absorbing $$\mathrm {C}^*$$ C ∗ -algebras

Unitary equivalence of automorphisms of separable C*-algebras

The complexity of the homeomorphism relation between compact metric spaces

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Turbulence, orbit equivalence, and the classification of nuclear C^* -algebras