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ERGODIC EXTENSIONS OF ENDOMORPHISMS

Part of: Selfadjoint operator algebras Individual linear operators as elements of algebraic systems Ergodic theory

Published online by Cambridge University Press:  02 October 2015

EVGENIOS T. A. KAKARIADIS  and
JUSTIN R. PETERS
EVGENIOS T. A. KAKARIADIS*
Affiliation:
School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK email evgenios.kakariadis@ncl.ac.uk
JUSTIN R. PETERS
Affiliation:
Department of Mathematics, Iowa State University, Ames, Iowa, IA 50011, USA email peters@iastate.edu
*
✉evgenios.kakariadis@ncl.ac.uk
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Abstract

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We examine a class of ergodic transformations on a probability measure space $(X,{\it\mu})$ and show that they extend to representations of ${\mathcal{B}}(L^{2}(X,{\it\mu}))$ that are both implemented by a Cuntz family and ergodic. This class contains several known examples, which are unified in our work. During the analysis of the existence and uniqueness of this Cuntz family, we find several results of independent interest. Most notably, we prove a decomposition of $X$ for $N$-to-one local homeomorphisms that is connected to the orthonormal bases of certain Hilbert modules.

Keywords

maximal abelian selfadjoint algebrasergodic transformationsHilbert modules

MSC classification

Primary: 47C15: Operators in $C^*$- or von Neumann algebras
Secondary: 37A55: Relations with the theory of $C^*$-algebras 46L55: Noncommutative dynamical systems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 93 , Issue 2 , April 2016 , pp. 307 - 320
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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