Almost Finiteness for General Étale Groupoids and Its Applications to Stable Rank of Crossed Products

Suzuki, Yuhei

doi:10.1093/imrn/rny187

Cited by 16 publications

(34 citation statements)

References 45 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Typical examples of free actions arise as the left translation actions associated to group embeddings into locally compact groups. Also, we have shown in [59] that (countable, non-torsion, exact) groups admit many minimal free actions on various compact spaces ( [59], Theorem B.1). Essentially free actions naturally appear as translation actions on homogeneous spaces and as Bernoulli shift actions; see [68].…”

Section: Introductionmentioning

confidence: 99%

Complete Descriptions of Intermediate Operator Algebras by Intermediate Extensions of Dynamical Systems

Suzuki

2019

Commun. Math. Phys.

Self Cite

View full text Add to dashboard Cite

Practically and intrinsically, inclusions of operator algebras are of fundamental interest. The subject of this paper is intermediate operator algebras of inclusions. There are two previously known theorems which naturally and completely describe all intermediate operator algebras: the Galois Correspondence Theorem and the Tensor Splitting Theorem. Here we establish the third, new complete description theorem which gives a canonical bijective correspondence between intermediate operator algebras and intermediate extensions of dynamical systems. One can also regard this theorem as a crossed product splitting theorem, analogous to the Tensor Splitting Theorem. We then give concrete applications, particularly to maximal amenability problem and a new realization result of intermediate operator algebra lattice.2000 Mathematics Subject Classification. Primary 46L05, 46L10, Secondary 46L55, 54H20.

show abstract

Section: Introductionmentioning

confidence: 99%

Complete Descriptions of Intermediate Operator Algebras by Intermediate Extensions of Dynamical Systems

Suzuki

2019

Commun. Math. Phys.

Self Cite

View full text Add to dashboard Cite

show abstract

“…As already mentioned in [28], every compact ample principal groupoid always admits a clopen castle by [20,Lemma 4.7]. It follows that Definition 1.2 is equivalent to the definition of almost finiteness given in [28,Definition 3.6] by Suzuki. Due to this fact, we will be using both equivalent notions of almost finiteness without further notice.…”

Section: (3)])mentioning

confidence: 86%

“…Definition 1.3. [28,Definition 3.2] Let K be a compact groupoid. A clopen castle for K is a partition…”

Section: (3)])mentioning

confidence: 99%

“…In [28] Suzuki develops a new strategy which in essence is a local version of Phillips' large subgroupoid technique ( [22]). Using this method, he is able to verify that almost finite minimal groupoid C*-algebras have stable rank 1.…”

mentioning

confidence: 99%

“…Using this method, he is able to verify that almost finite minimal groupoid C*-algebras have stable rank 1. Moreover, in [28,Remark 4.3] he claims that a suitably adapted strategy indeed also yields real rank zero and strict comparison for such groupoid C * -algebras. In this note we carry out all the necessary intermediate steps (some of which might be of independent interest) in detail, as we believe concrete proofs of these facts would be a useful contribution to the literature.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Strict comparison for $$C^*$$-algebras arising from almost finite groupoids

Ara

Bönicke

Bosa

et al. 2020

Banach J. Math. Anal.

View full text Add to dashboard Cite

In this paper we show that for an almost finite minimal ample groupoid G, its reduced C * -algebra C * r (G) has real rank zero and strict comparison even though C * r (G) may not be nuclear in general. Moreover, if we further assume G being also second countable and non-elementary, then its Cuntz semigroup Cu(C * r (G)) is almost divisible and Cu(C * r (G)) and Cu(C * r (G) ⊗ Z) are canonically order-isomorphic, where Z denotes the Jiang-Su algebra.Almost finiteness for an ample groupoid was introduced by Matui in [20]. He studied their topological full groups as well as the applications of almost finiteness to the homology of étale groupoids (see [21] for a survey of results in this direction). In [12], David Kerr specialised to almost finite group actions and treated them as a topological analogue of probability measure preserving hyperfinite equivalence relations, with the ultimate goal of transferring ideas from the classification of equivalence relations and von Neumann algebras to the world of (amenable) topological dynamics and C * -algebras.Recently, the classification program for C * -algebras has culminated in the outstanding theorem that all separable, simple, unital, nuclear, Z-stable C * -algebras satisfying the universal coefficient theorem (UCT) are classified by their Elliott-invariant (see [30, Corollary D] and [8, Corollary D]). Recall that a C * -algebra is Z-stable if A ⊗ Z ∼ = A, where Z denotes the so-called Jiang-Su algebra. By the Toms-Winter conjecture Z-stability is conjecturally equivalent to strict comparison (or equivalently, almost unperforation of the Cuntz semigroup) for separable, simple, nuclear, non-elementary C * -algebras. It is known that Z-stability implies strict comparison in general and the converse is indeed the last remaining open step in the Toms-Winter conjecture (see [33] for an overview and [7] for the state of the art for the conjecture).Going back to topological dynamics, David Kerr's approach in [12] has seen dramatic success. He was able to show that a crossed product C(X) Γ associated to a free and minimal action of an (amenable) infinite group Γ is Z-stable provided that the action is almost finite (see [12, Theorem 12.4]). Combining this with the recent result in [13, Theorem 8.1], which states that every free action of a countably infinite (amenable) group with subexponential

show abstract

The type semigroup, comparison, and almost finiteness for ample groupoids

Ara

Bönicke

Bosa³

et al. 2021

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

We prove that a minimal second countable ample groupoid has dynamical comparison if and only if its type semigroup is almost unperforated. Moreover, we investigate to what extent a not necessarily minimal almost finite groupoid has an almost unperforated type semigroup. Finally, we build a bridge between coarse geometry and topological dynamics by characterizing almost finiteness of the coarse groupoid in terms of a new coarsely invariant property for metric spaces, which might be of independent interest in coarse geometry. As a consequence, we are able to construct new examples of almost finite principal groupoids lacking other desirable properties, such as amenability or even a-T-menability. This behaviour is in stark contrast to the case of principal transformation groupoids associated to group actions.

show abstract

Almost Finiteness for General Étale Groupoids and Its Applications to Stable Rank of Crossed Products

Cited by 16 publications

References 45 publications

Complete Descriptions of Intermediate Operator Algebras by Intermediate Extensions of Dynamical Systems

Complete Descriptions of Intermediate Operator Algebras by Intermediate Extensions of Dynamical Systems

Strict comparison for $$C^*$$-algebras arising from almost finite groupoids

The type semigroup, comparison, and almost finiteness for ample groupoids

Contact Info

Product

Resources

About