Se p 20 07 Operator algebras for multivariable dynamics

Let X be a locally compact Hausdorff space with n proper continuous self maps σi : X → X for 1 ≤ i ≤ 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 A(X, τ) and the semicrossed product C0(X)×τ Fn . 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 A(X, σ) or C0(X)×σ Fn . 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 A(X, σ) or C0(X) ×σ Fn is semisimple if and only if there are no generalized wandering sets. In the metrizable case, this is equivalent to each σi being surjective and v-recurrent points being dense for each v ∈ Fn . 2000 Mathematics Subject Classification. Primary 47L55; Secondary 47L40, 46L05, 37B20, 37B99.