scholarly journals Skew-product dynamical systems for crossed product C⁎-algebras and their ergodic properties

2021 ◽  
pp. 125302
Author(s):  
Simone Del Vecchio ◽  
Francesco Fidaleo ◽  
Stefano Rossi
Keyword(s):  
Dynamical Systems ◽  
Crossed Product ◽  
Skew Product ◽  
C Algebras ◽  
Ergodic Properties

The Local Product Structure of Expansive Automorphisms of Solenoids and Their Associated C*-Algebras

1996 ◽  
Vol 48 (4) ◽  
pp. 692-709 ◽  
Author(s):  
Berndt Brenken
Keyword(s):  
Abelian Group ◽  
Dynamical Systems ◽  
Coordinate System ◽  
Finite Type ◽  
Crossed Product ◽  
C Algebras ◽  
Local Product ◽  
Subshifts Of Finite Type ◽  
Canonical Coordinate System ◽  
Canonical Coordinate

AbstractAn explicit description of a hyperbolic canonical coordinate system for an expansive automorphism of a compact connected abelian group is given. These dynamical systems are factors of subshifts of finite type. Some properties of the associated crossed product C*-algebra are discussed. In these examples, the C* -algebras of Ruelle are crossed product algebras.

scholarly journals Ergodic properties of the Anzai skew-product for the non-commutative torus

2020 ◽  
pp. 1-22 ◽  
Author(s):  
SIMONE DEL VECCHIO ◽  
FRANCESCO FIDALEO ◽  
LUCA GIORGETTI ◽  
STEFANO ROSSI
Keyword(s):  
Dynamical Systems ◽  
Unit Circle ◽  
Systematic Study ◽  
Cartesian Product ◽  
Skew Product ◽  
Quantum Dynamical ◽  
Ergodic Properties ◽  
Pointwise Limit ◽  
Quantum Dynamical Systems ◽  
Uniquely Ergodic

We provide a systematic study of a non-commutative extension of the classical Anzai skew-product for the cartesian product of two copies of the unit circle to the non-commutative 2-tori. In particular, some relevant ergodic properties are proved for these quantum dynamical systems, extending the corresponding ones enjoyed by the classical Anzai skew-product. As an application, for a uniquely ergodic Anzai skew-product $\unicode[STIX]{x1D6F7}$ on the non-commutative $2$ -torus $\mathbb{A}_{\unicode[STIX]{x1D6FC}}$ , $\unicode[STIX]{x1D6FC}\in \mathbb{R}$ , we investigate the pointwise limit, $\lim _{n\rightarrow +\infty }(1/n)\sum _{k=0}^{n-1}\unicode[STIX]{x1D706}^{-k}\unicode[STIX]{x1D6F7}^{k}(x)$ , for $x\in \mathbb{A}_{\unicode[STIX]{x1D6FC}}$ and $\unicode[STIX]{x1D706}$ a point in the unit circle, and show that there are examples for which the limit does not exist, even in the weak topology.

scholarly journals Exel's crossed product for non-unital C*-algebras

2010 ◽  
Vol 149 (3) ◽  
pp. 423-444 ◽  
Author(s):  
NATHAN BROWNLOWE ◽  
IAIN RAEBURN ◽  
SEAN T. VITTADELLO
Keyword(s):  
Dynamical Systems ◽  
Transfer Operator ◽  
Finite Graph ◽  
Crossed Product ◽  
Crossed Products ◽  
Ideal Structure ◽  
Locally Finite ◽  
C Algebras ◽  
Locally Finite Graph ◽  
The Ideal

AbstractWe consider a family of dynamical systems (A, α, L) in which α is an endomorphism of a C*-algebra A and L is a transfer operator for α. We extend Exel's construction of a crossed product to cover non-unital algebras A, and show that the C*-algebra of a locally finite graph can be realised as one of these crossed products. When A is commutative, we find criteria for the simplicity of the crossed product, and analyse the ideal structure of the crossed product.

scholarly journals Classifying crossed product C*-algebras

2016 ◽  
Vol 138 (3) ◽  
pp. 793-820 ◽  
Author(s):  
Wilhelm Winter
Keyword(s):  
Crossed Product ◽  
C Algebras

CROSSED PRODUCT C*-ALGEBRAS AND ALGEBRAIC TOPOLOGY

1996 ◽  
Vol 08 (04) ◽  
pp. 623-637
Author(s):  
JUDITH A. PACKER
Keyword(s):  
Algebraic Topology ◽  
Equivariant Cohomology ◽  
Automorphism Groups ◽  
Crossed Product ◽  
Fiber Bundles ◽  
Topological Invariants ◽  
C Algebras ◽  
Recent Developments ◽  
Continuous Trace ◽  
Cech Cohomology

We discuss some recent developments that illustrate the interplay between the theory of crossed products of continuous trace C*-algebras and algebraic topology, summarizing results relating topological invariants coming from the theory of fiber bundles to continuous trace C*-algebras and their automorphism groups and the structure of the associated crossed product C*-algebras. This survey article starts from the classical theory of Dixmier, Douady, and Fell, and discusses the more recent work of Echterhoff, Phillips, Raeburn, Rosenberg, and Williams, among others. The topological invariants involved are Čech cohomology, the cohomology of locally compact groups with Borel cochains of C. Moore, and the recently introduced equivariant cohomology theory of Crocker, Kumjian, Raeburn and Williams.

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