Isomorphisms between Topological Conjugacy Algebras

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 ηi : Xi → Xi is a continuous proper map on a locally compact Hausdorff space Xi, for i = 1, 2. We show that the dynamical systems (X1, η1) and (X2, η2) are conjugate if and only if some topological conjugacy algebra of (X1, η1) is isomorphic as an algebra to some topological conjugacy algebra of (X2, η2). This implies as a corollary the complete classification of the semicrossed products C0(X ) ×η Z , which was previously considered by Arveson and Josephson [3], Peters [19], Hadwin and Hoover [8] and Power [21]. We also obtain a complete classification of all semicrossed products of the form A(D) ×η Z , where A(D) denotes the disc algebra and η : D → D 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.