Cartan subalgebras in C*-algebras. Existence and uniqueness

Li, Xin; Renault, Jean

doi:10.1090/tran/7654

Cited by 51 publications

(24 citation statements)

References 71 publications

(133 reference statements)

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“…As an immediate consequence of Proposition 2.2, we obtain the following corollary. This strengthens and unifies Corollaries 4.7 and 4.11 in [18] (cf. Corollary B of [21], Theorem B of [3]).…”

Section: Tensorial Primeness a Von Neumann Algebrasupporting

confidence: 79%

“…Besides from known facts, we include a new general statement on finite GNS-completions of regular abelian * -subalgebras (Proposition 2.2). This in particular strengthens and unifies Corollaries 4.7 and 4.11 of [18] (see Corollary 2.3). In Section 3, we introduce and study iterated wreath product groups.…”

Section: Introductionsupporting

confidence: 78%

“…This implies a C * -algebra analogue of Lemma 2.1. We note that the last statement is shown in Lemma 4.10 of [18] (see also Corollary 4.11 in [18]) for Cantan subalgebras in separable C * -algebras. We point out that our proof does not rely on the groupoid realization theorem [25] unlike their proof.…”

Section: Tensorial Primeness a Von Neumann Algebramentioning

confidence: 54%

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Rigid Sides of Approximately Finite-Dimensional Simple Operator Algebras in Non-Separable Category

Suzuki

2019

International Mathematics Research Notices

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Applying Popa's orthogonality method to a new class of groups, we construct amenable group factors which are prime and have no infinite dimensional regular abelian * -subalgebras. By adjusting Farah-Katsura's solution of Dixmier's problem to the von Neumann algebra setting, we obtain the first examples of prime AFD factors and tensorially prime simple AF-algebras. Our results are proved in ZFC, thus in particular answering questions asked by Farah-Hathaway-Katsura-Tikuisis. We also directly determine central sequences of certain crossed products. This concludes the failure of the Kirchberg O ∞ -absorption theorem in the non-separable setting.2000 Mathematics Subject Classification. Primary 46L05, 46L10.

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Section: Tensorial Primeness a Von Neumann Algebrasupporting

confidence: 79%

Section: Introductionsupporting

confidence: 78%

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Rigid Sides of Approximately Finite-Dimensional Simple Operator Algebras in Non-Separable Category

Suzuki

2019

International Mathematics Research Notices

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“…Remark 2.22. In combination with [44], Corollary 2.21 implies that nuclear Roe algebras have distinguished Cartan subalgebras, as explained in [28].…”

Section: Dynamic Characterizations Of Coarse Embeddings Equivalencesmentioning

confidence: 92%

Dynamic characterizations of quasi-isometry and applications to cohomology

2018

Algebr. Geom. Topol.

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We build a bridge between geometric group theory and topological dynamical systems by establishing a dictionary between coarse equivalence and continuous orbit equivalence. As an application, we give conceptual explanations for previous results of Shalom and Sauer on coarse invariance of homological and cohomological dimensions and Shalom's property H FD . As another application, we show that group homology and cohomology in a class of coefficients, including all induced and co-induced modules, are coarse invariants. We deduce that being of type FP n (over arbitrary rings) is a coarse invariant, and that being a (Poincaré) duality group over a ring is a coarse invariant among all groups which have finite cohomological dimension over that ring. Our results also imply that every self coarse embedding of a Poincaré duality group over an arbitrary ring must be a coarse equivalence. It is then evident that coarse equivalences induce isomorphisms as they are precisely those coarse maps which are invertible modulo ∼. A similar remark applies to cohomology. Thus, not only these (co)homology groups, but, by functoriality, the actions of the groups of coarse equivalences (modulo ∼) on these (co)homology groups are coarse invariants as well. We obtain analogous results for coarse embeddings in the topological setting, i.e., for topological res-invariant modules and reduced (co)homology. It turns out that coarse embeddings always induce isomorphisms in (co)homology and reduced (co)homology.The aforementioned results on coarse invariance of type FP n and being a (Poincaré) duality group are immediate consequences, as is our rigidity result for coarse embeddings into Poincaré duality groups. We also deduce that vanishing of ℓ 2 -Betti numbers is a coarse invariant, as observed in [35,34,32], and generalized by Sauer and Schrödl to all unimodular locally compact second countable groups [42]. This is a good point to formulate an interesting and natural question, which we elaborate on in § 4.4:Question (Question 4.44). Are homological and cohomological dimension over a commutative ring R with unit always coarse invariants among all countable discrete groups with no R-torsion?We refer to § 4 for more details. § 3 and § 4 are independent from each other. Thus readers interested in this last set of results on coarse invariance of group (co)homology may go directly from § 2 to § 4.As far as our methods are concerned, we use groupoid techniques as in [43,41,34], but instead of working with abstract dynamical systems, we base our work on concrete dynamic characterizations of coarse equivalence. The difference between our work and [19] is that we do not work with descriptions of group (co)homology in terms of Eilenberg-MacLane spaces, as these descriptions require finiteness conditions (like F n or F ∞ ) on our groups and have to be modified whenever we change coefficients. Instead, since coarse embeddings automatically lead to "controlled" orbit equivalences satisfying the finiteness condition mentioned above, we can work directly with...

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“…Finally let τ := τ 0 • E, where E : C * r (G) → C(G (0) ) is the canonical faithful conditional expectation. It is clear, that τ is a faithful state and the trace property follows from the G-invariance of τ 0 (see [33,Lemma 4.2] for a recent proof of this fact).…”

Section: As Follows: For Setsmentioning

confidence: 99%

Ideal structure and pure infiniteness of ample groupoid -algebras

Bönicke¹,

Li²

2018

Ergod. Th. Dynam. Sys.

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In this paper, we study the ideal structure of reduced C * -algebras C * r (G) associated to étale groupoids G. In particular, we characterize when there is a one-to-one correspondence between the closed, two-sided ideals in C * r (G) and the open invariant subsets of the unit space G (0) of G. As a consequence, we show that if G is an inner exact, essentially principal, ample groupoid, then C * r (G) is (strongly) purely infinite if and only if every non-zero projection in C0(G (0) ) is properly infinite in C * r (G). We also establish a sufficient condition on the ample groupoid G that ensures pure infiniteness of C * r (G) in terms of paradoxicality of compact open subsets of the unit space G (0) .Finally, we introduce the type semigroup for ample groupoids and also obtain a dichotomy result: Let G be an ample groupoid with compact unit space which is minimal and topologically principal. If the type semigroup is almost unperforated, then C * r (G) is a simple C * -algebra which is either stably finite or strongly purely infinite.

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Cartan subalgebras in C*-algebras. Existence and uniqueness

Cited by 51 publications

References 71 publications

Rigid Sides of Approximately Finite-Dimensional Simple Operator Algebras in Non-Separable Category

Rigid Sides of Approximately Finite-Dimensional Simple Operator Algebras in Non-Separable Category

Dynamic characterizations of quasi-isometry and applications to cohomology

Ideal structure and pure infiniteness of ample groupoid -algebras

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