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Operator algebras for multivariable dynamics
About this Title
Kenneth R. Davidson, Pure Mathematics Department, University of Waterloo, Waterloo, Ontario N2L–3G1, Canada and Elias G. Katsoulis, Department of Mathematics, East Carolina University, Greenville, North Carolina 27858
Publication: Memoirs of the American Mathematical Society
Publication Year:
2011; Volume 209, Number 982
ISBNs: 978-0-8218-5302-3 (print); 978-1-4704-0596-0 (online)
DOI: https://doi.org/10.1090/S0065-9266-10-00615-0
Published electronically: June 8, 2010
Keywords: multivariable dynamical system,
operator algebra,
tensor algebra,
semi-crossed product,
Cuntz-Pimsner C*-algebra,
semisimple,
radical,
piecewise conjugacy,
wandering sets,
recurrence
MSC: Primary 47L55; Secondary 47L40, 46L05, 37B20, 37B99
Table of Contents
Chapters
- 1. Introduction
- 2. Dilation Theory
- 3. Recovering the Dynamics
- 4. Semisimplicity
- 5. Open Problems and Future Directions
Abstract
Let $X$ be a locally compact Hausdorff space with $n$ proper continuous self maps $\sigma _i:X \to X$ for $1 \le i \le n$. To this we associate two conjugacy operator algebras which emerge as the natural candidates for the universal algebra of the system, the tensor algebra $\mathcal {A}(X,\tau )$ and the semicrossed product $\mathrm {C}_0(X)\times _\tau \mathbb {F}_n^+$.
We develop the necessary dilation theory for both models. In particular, we exhibit an explicit family of boundary representations which determine the C*-envelope of the tensor algebra.
We introduce a new concept of conjugacy for multidimensional systems, called piecewise conjugacy. We prove that the piecewise conjugacy class of the system can be recovered from the algebraic structure of either $\mathcal A( X , \sigma )$ or $\mathrm {C}_0(X)\times _\sigma \mathbb {F}_n^+$. Various classification results follow as a consequence. For example, if $n=2$ or $3$, or the space $X$ has covering dimension at most 1, then the tensor algebras are algebraically isomorphic (or completely isometrically isomorphic) if and only if the systems are piecewise topologically conjugate.
We define a generalized notion of wandering sets and recurrence. Using this, it is shown that $\mathcal A( X , \sigma )$ or $\mathrm {C}_0(X)\times _\sigma \mathbb {F}_n^+$ is semisimple if and only if there are no generalized wandering sets. In the metrizable case, this is equivalent to each $\sigma _i$ being surjective and $v$-recurrent points being dense for each $v \in \mathbb {F}_n^+$.
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