Cartan Subalgebras in $C^*$-Algebras

Renault, Jean

doi:10.33232/bims.0061.29.63

Cited by 253 publications

(376 citation statements)

References 58 publications

Supporting

Mentioning

368

Contrasting

Order By: Relevance

“…The equivalence of (ii) and (iii) follows from Proposition 4.11. From Lemma 4.6, R ⋊ α G and S ⋊ β H are topological principal, thus the equivalence of (iii) and (iv) follows from [7,22].…”

Section: By a Similar Way We Can Show That A(ϕmentioning

confidence: 91%

“…As a topological analogue of the classification results on the probability measure preserving actions in the sense of orbit equivalence, Li introduced the notion of continuous orbit equivalence for continuous group actions and proved that two topologically free systems are continuously orbit equivalent if and only if their associated transformation groupoids are isomorphic ( [10]). By Renault's result in [22], these conditions are also equivalent to the existence of C * -isomorphism preserving the canonical Cartan subalgebras between the corresponding crossed product algebras. In [7], Li's rigidity result is generalized to the case of group actions with torsion-free and abelian essential stabilisers.…”

Section: Introductionmentioning

confidence: 90%

See 1 more Smart Citation

Continuous orbit equivalence up to equivalence relations

Qiang¹,

Hou²

2021

Preprint

View full text Add to dashboard Cite

We introduce notions of continuous orbit equivalence and strong (respective, weak) continuous orbit equivalence for automorphism systems of étale equivalence relations, and characterize them in terms of the semi-direct product groupoids, as well as their reduced groupoid C * -algebras with canonical Cartan subalgebras. In particular, we study topological rigidity of expansive automorphism actions on compact (connected) metrizable groups.

show abstract

Section: By a Similar Way We Can Show That A(ϕmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 90%

Continuous orbit equivalence up to equivalence relations

Qiang¹,

Hou²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…As shown in the proof of Lemma 7.6, G is not principal but is topologically free (or topologically principal ). By [24,Proposition 5.11], (C * (G), C(G (0) )) is a Cartan pair not having the unique extension property, whereas (A θ , D θ ) is a Cartan pair that does have that property. Therefore C(G (0) ) and D θ are nonconjugate Cartan subalgebras in A θ ∼ = C * (G).…”

Section: Identifying C * (G)mentioning

confidence: 99%

AF $C^*$-algebras from non AF groupoids

Mitscher¹,

Spielberg²

2021

Preprint

View full text Add to dashboard Cite

We construct ample groupoids from certain categories of paths, and prove that their C * -algebras coincide with the continued fraction AF algebras of Effros and Shen. The proof relies on recent classification results for simple nuclear C * -algebras. The groupoids are not principal. This provides examples of Cartan subalgebras in the continued fraction AF algebras that are isomorphic, but not conjugate, to the standard diagonal subalgebras.

show abstract

“…An action (X, Γ, Φ) is said to be free if for all x ∈ X and g ∈ Γ, g • x = x implies that g = e, the identity of the group. The notion of a topologically free action is a generalization of free actions, introduced by Boyle in his thesis [11], and later used in the works by Boyle and Tomiyama [12] for the study of classification of general Cantor actions, by Renault [43] for the study of the C * -algebras associated to Cantor actions, and by Li [37] for proving rigidity properties of Cantor actions. We recall this definition.…”

Section: Regularity Of Cantor Actionsmentioning

confidence: 99%

The prime spectrum of solenoidal manifolds

Hurder¹,

Lukina²

2021

Preprint

View full text Add to dashboard Cite

A solenoidal manifold is the inverse limit space of a tower of proper coverings of a compact manifold. In this work, we introduce new invariants for solenoidal manifolds, their asymptotic Steinitz orders and their prime spectra, and show they are invariants of the homeomorphism type. These invariants are formulated in terms of the monodromy Cantor action associated to a solenoidal manifold. To this end, we continue our study of invariants for minimal equicontinuous Cantor actions. We introduce the three types of prime spectra associated to such actions, and study their invariance properties under return equivalence. As an application, we show that a nilpotent Cantor action with finite prime spectrum must be stable. Examples of stable actions of the integer Heisenberg group are given with arbitrary prime spectrum. We also give the first examples of nilpotent Cantor actions which are wild, and not stable. Contents 1. Introduction 2. Cantor actions 2.1. Basic concepts 2.2. Equivalence of Cantor actions 2.3. Morita equivalence 2.4. Regularity of Cantor actions 3. Steinitz orders of Cantor actions 3.1. Abstract Steinitz orders 3.2. Orders and return equivalence 3.3. Algebraic model 3.4. Steinitz orders for algebraic models 3.5. Steinitz orders of solenoidal manifolds 4. Nilpotent actions 4.1. Noetherian groups 4.2. Dynamics of Noetherian groups 5. Examples 5.1. Toroidal actions 5.2. Heisenberg actions References

show abstract

Cartan Subalgebras in $C^*$-Algebras

Cited by 253 publications

References 58 publications

Continuous orbit equivalence up to equivalence relations

Continuous orbit equivalence up to equivalence relations

AF $C^*$-algebras from non AF groupoids

The prime spectrum of solenoidal manifolds

Contact Info

Product

Resources

About