Cuntz-like algebras

Abstract

The usual crossed product construction which associates to the homeomorphism $T$ of the locally compact space $X$ the C${}^{\ast }$-algebra $C^{\ast }(X,T)$ is extended to the case of a partial local homeomorphism $T$. For example, the Cuntz-Krieger algebras are the C${}^{\ast }$-algebras of the one-sided Markov shifts. The generalizations of the Cuntz-Krieger algebras (graph algebras, algebras $O_A$ where $A$ is an infinite matrix) which have been introduced recently can also be described as C${}^{\ast }$-algebras of Markov chains with countably many states. This is useful to obtain such properties of these algebras as nuclearity, simplicity or pure infiniteness. One also gives examples of strong Morita equivalences arising from dynamical systems equivalences.

18 pages, AMS LaTeX, to appear in the Proceedings of the 17th Conference in Operator Theory at Timisoara

Cited by in corpus (13)

• Shift tail equivalence and an unbounded representative of the Cuntz-Pimsner extension
• Conjugacy of local homeomorphisms via groupoids and C*-algebras
• KMS states on C^*-algebras associated with self-similar sets
• C*-algebras associated with branched coverings
• Topological full groups of ultragraph groupoids as an isomorphism invariant
• A note on the core of Steinberg algebras
• Multiply generated dynamical systems and the duality of higher rank graph algebras
• Quasi-invariant measures for generalized approximately proper equivalence relations
• Groupoid Actions on Fractafolds
• KMS states on C*-algebras associated to expansive maps
• Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States
• Operator Algebras and Mauldin Williams Graphs
• Cocycles on Certain Groupoids Associated to ${\mathbb{N}}^k$-Actions