arXiv subjects
Publications and source records attributed to Elias G. Katsoulis.
Let $(\A, α)$ and $(\B, β)$ be C*-dynamical systems and assume that $\A$ is a separable simple C*-algebra and that $α$ and $β$ are *-automorphisms. Then the semicrossed products $\A \times_α \bbZ^{+}$ and $\B \times_β \bbZ^{+}$ are isometrically isomorphic if and only if the dynamical systems $(\A, α)$ and $(\B, β)$ are outer conjugate.
A family of algebras, which we call topological conjugacy algebras, is associated with each proper continuous map on a locally compact Hausdorff space. Assume that $η_i:\X_i\to \X_i$ is a continuous proper map on a locally compact Hausdorff space $\X_i$, for $i = 1,2$. We show that the dynamical systems $(\X_1, η_1)$ and $(\X_2, η_2)$ are conjugate if and only if some topological conjugacy algebra of $(\X_1, η_1)$ is isomorphic as an algebra to some topological conjugacy algebra of $(\X_2, η_2)$. This implies as a corollary the complete classification of the semicrossed products $C_0(\X) \times_η \bbZ^{+}$, which was previously considered by Arveson and Josephson, Peters, Hadwin and Hoover and Power. We also obtain a complete classification of all semicrossed products of the form $A(\bbD) \times_η\bbZ^{+}$, where $A(\bbD)$ denotes the disc algebra and $η: \bbD \to \bbD$ a continuous map which is analytic on the interior. In this case, a surprising dichotomy appears in the classification scheme, which depends on the fixed point set of $η$. We also classify more general semicrossed products of uniform algebras.
Let $X$ be a locally compact Hausdorff space with $n$ proper continuous self maps $τ_i:X \to X$ 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 $\A(X, τ)$ and the semicrossed product $\rC_0(X)\times_τ\Fn$. 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 $\A(X, τ)$ or $\rC_0(X)\times_τ\Fn$. 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 $\A(X, τ)$ or $\rC_0(X)\times_τ\Fn$ is semisimple if and only if there are no generalized wandering sets. In the metrizable case, this is equivalent to each $τ_i$ being surjective and $v$-recurrent points being dense for each $v \in \Fn$.
We show that the $\ca$-envelope of the tensor algebra of an arbitrary $\ca$-correspondence $\X$ coincides with the Cuntz-Pimsner algebra $Ø_{\X}$, as defined by Katsura \cite{Ka}. This improves earlier results of Muhly and Solel and Fowler, Muhly and Raeburn, who came to the same conclusion under the additional hypothesis that $\X$ is strict and faithful.
We prove that algebraic isomorphisms between limit algebras are automatically continuous, and consider consequences of this result. In particular, we give partial solutions to a conjecture of Power [Limit Algebras, Longman, 1992, Notes to Chapter 8] and to an open problem [op. cit., Problem 7.8]. As a further consequence, we describe epimorphisms between various classes of limit algebras.