Group actions on Smale space -algebras

Deeley, Robin J.; Strung, Karen R.

doi:10.1017/etds.2019.11

Cited by 3 publications

(3 citation statements)

References 64 publications

(107 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To prove (14), we use the third part of (3.16), (3.17), (3.21), and (3.15) at the last step to estimate…”

Section: Lemma 34 Let a Be A C*-algebra And Letmentioning

confidence: 99%

“…We use (14) at the first step and use (4) at the third step to get 2) and (3) at the second step, and using (1) at the third step, we get…”

Section: Lemma 34 Let a Be A C*-algebra And Letmentioning

confidence: 99%

“…The importance of this class of group actions goes beyond this and will appear in [8]. Further, one may even study the notion weak tracial approximate representability on Smale space C*-algebras and investigate some results of [14] by considering this class of group actions.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Weakly tracially approximately representable actions

Asadi-Vasfi

2021

Preprint

View full text Add to dashboard Cite

We describe a weak tracial analog of approximate representability under the name "weak tracial approximate representability" for finite group actions. We then investigate the dual actions on the crossed products by this class of group actions. Namely, let G be a finite abelian group, let A be an infinite-dimensional simple unital C*-algebra, and let α : G → Aut(A) be an action of G on A which is pointwise outer. Then α has the weak tracial Rokhlin property if and only if the dual action α of the Pontryagin dual G on the crossed product C * (G, A, α) is weakly tracially approximately representable, and α is weakly tracially approximately representable if and only if the dual action α has the weak tracial Rokhlin property. This generalizes the results of Izumi in 2004 and Phillips in 2011 on the dual actions of finite abelian groups on unital simple C*-algebras.We further obtain some results on the Cuntz semigroup and the radius of comparison of the crossed product by a finite group action which has a weakened version of weak tracial approximate representability. Namely, let G be a finite group, let A be an infinite-dimensional stably finite simple unital C*algebra, and let α : G → Aut(A) be a weakly tracially strictly approximately inner action of G on A. Let A α be the fixed point algebra. Then the radius of comparison satisfies rc(A) ≤ rc C * (G, A, α) and if C * (G, A, α) is simple, then rc(A) ≤ rc C * (G, A, α) ≤ rc(A α ). Further, the inclusion of A in C * (G, A, α) induces an isomorphism from the soft part of the Cuntz semigroup Cu(A) to its image in Cu(C * (G, A, α)). These results generalize our earlier results on the radius of comparison and the Cuntz semigroup of the crossed product by a tracially strictly approximately inner action.

show abstract

“…To prove (14), we use the third part of (3.16), (3.17), (3.21), and (3.15) at the last step to estimate…”

Section: Lemma 34 Let a Be A C*-algebra And Letmentioning

confidence: 99%

“…We use (14) at the first step and use (4) at the third step to get 2) and (3) at the second step, and using (1) at the third step, we get…”

Section: Lemma 34 Let a Be A C*-algebra And Letmentioning

confidence: 99%

See 1 more Smart Citation

Weakly tracially approximately representable actions

Asadi-Vasfi

2021

Preprint

View full text Add to dashboard Cite

show abstract

Binary factors of shifts of finite type

Putnam

2023

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

We construct two new classes of topological dynamical systems; one is a factor of a one-sided shift of finite type while the second is a factor of the two-sided shift. The data are a finite graph which presents the shift of finite type, a second finite directed graph and a pair of embeddings of it into the first, satisfying certain conditions. The factor is then obtained from a simple idea based on binary expansion of real numbers. In both cases, we construct natural metrics on the factors and, in the second case, this makes the system a Smale space, in the sense of Ruelle. We compute various algebraic invariants for these systems, including the homology for Smale space developed by the author and the K-theory of various $C^{*}$ -algebras associated to them, in terms of the pair of original graphs.

show abstract

On finitely summable Fredholm modules from Smale spaces

Gerontogiannis

2022

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

We prove that all K K -homology classes of the stable (and unstable) Ruelle algebra of a Smale space have explicit Fredholm module representatives that are finitely summable on the same smooth subalgebra and with the same degree of summability. The smooth subalgebra is induced by a metric on the underlying Smale space groupoid and fine transversality relations between stable and unstable sets. The degree of summability is related to the fractal dimension of the Smale space. Further, the Fredholm modules are obtained by taking Kasparov products with a fundamental class of the Spanier-Whitehead K K -duality between the Ruelle algebras. Finally, we obtain general results on stability under holomorphic functional calculus and construct Lipschitz algebras on étale groupoids.

show abstract

Group actions on Smale space -algebras

Cited by 3 publications

References 64 publications

Weakly tracially approximately representable actions

Weakly tracially approximately representable actions

Binary factors of shifts of finite type

On finitely summable Fredholm modules from Smale spaces

Contact Info

Product

Resources

About