Operator algebras for multivariable dynamics

Details Operator algebras for multivariable dynamics Operator algebras for multivariable dynamics

2007-01-18

arxiv.org/abs/math/0701514v5

Mathematics

Operator Algebras

In this version we explicitly exhibit enough (irreducible) boundary repns to faithfully represent the C*-envelope of the tenso

Scientific paper

Let $X$ be a locally compact Hausdorff space with $n$ proper continuous self maps $\tau_i:X \to X$ for $1 \le i \le n$. To this we associate two topological conjugacy algebras which emerge as the natural candidates for the universal algebra of the system, the tensor algebra $\A(X, \tau)$ and the semicrossed product $\rC_0(X)\times_\tau\Fn$. We introduce a concept of conjugacy for multidimensional systems, which we coin piecewise conjugacy. We prove that the piecewise conjugacy class of the system can be recovered from either the algebraic structure of $\A(X, \tau)$ or $\rC_0(X)\times_\tau\Fn$. Various classification results follow as a consequence. For example, for $n=2,3$, the tensor algebras are (algebraically or even completely isometrically) isomorphic if and only if the systems are piecewise topologically conjugate. In order to establish these results we make use of analytic varieties as well as homotopy theory for Lie groups We define a generalized notion of wandering sets and recurrence. Using this, it is shown that $\A(X, \tau)$ or $\rC_0(X)\times_\tau\Fn$ is semisimple if and only if there are no generalized wandering sets. In the metrizable case, this is equivalent to each $\tau_i$ being surjective and $v$-recurrent points being dense for each $v \in \Fn$.

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