A note on the ideals of groupoid <i>C</i>*-algebras from Smale spaces

Hou, Chengjun; Chen, Xiamoman

doi:10.1017/s0004972700039939

Cited by 4 publications

(9 citation statements)

References 4 publications

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“…Hence if log(λ) is the entropy then λ > 1. (2) The homoclinic algebra of a mixing Smale space has a unique tracial state τ H , see [16] and Subsection 2.5 above.…”

Section: Real Rank Zeromentioning

confidence: 99%

Smale space $C^*$-algebras have nonzero projections

Deeley¹,

Goffeng²,

Yashinski³

2019

Proc. Amer. Math. Soc.

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The main result of the present paper is that the stable and unstable C * -algebras associated to a mixing Smale space always contain nonzero projections. This gives a positive answer to a question of the first listed author and Karen Strung and has implications for the structure of these algebras in light of the Elliott program for simple C * -algebras. Using our main result, we also show that the homoclinic, stable, and unstable algebras each have real rank zero.

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“…Hence if log(λ) is the entropy then λ > 1. (2) The homoclinic algebra of a mixing Smale space has a unique tracial state τ H , see [16] and Subsection 2.5 above.…”

Section: Real Rank Zeromentioning

confidence: 99%

Smale space $C^*$-algebras have nonzero projections

Deeley¹,

Goffeng²,

Yashinski³

2019

Proc. Amer. Math. Soc.

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“…The traces on C * (S) and C * (U ) are not bounded while C * (H) admits a tracial state. Moreover, when (X, ϕ) is mixing, this trace is unique [25].…”

Section: Definitionmentioning

confidence: 99%

Group actions on Smale space -algebras

Deeley¹,

Strung²

2019

Ergod. Th. Dynam. Sys.

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Group actions on a Smale space and the actions induced on the C * -algebras associated to such a dynamical system are studied. We show that an effective action of a discrete group on a mixing Smale space produces a strongly outer action on the homoclinic algebra. We then show that for irreducible Smale spaces, the property of finite Rokhlin dimension passes from the induced action on the homoclinic algbera to the induced actions on the stable and unstable C * -algebras. In each of these cases, we discuss the preservation of properties-such as finite nuclear dimension, Z-stability, and classification by Elliott invariants-in the resulting crossed products.

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“…Abstract and Applied Analysis 3 Then, is a second countable, locally compact, Hausdorff,discrete, and principal groupoid under this topology, and the counting measure is a Haar system [2,7,8]. The path space Λ Δ is identified with the unit space 0 by the embedding map → ( , ).…”

Section: 2mentioning

confidence: 99%

“…For an irreducible Smale space ( , ), Putnam and Spielberg [3] showed that groupoids defined by stable and unstable equivalence relations are essentially principal, and Hou and Chen [7] showed that ⋊ Z is essentially principal. Recall that a Smale space is said to be irreducible if it is nonwandering and has a dense orbit.…”

Section: Propositionmentioning

confidence: 99%