Abstract

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 is a continuous proper map on a locally compact Hausdorff space , for i= 1,2. We show that the dynamical systems and are conjugate if and only if some topological conjugacy algebra of is isomorphic as an algebra to some topological conjugacy algebra of . This implies as a corollary the complete classification of the semicrossed products , which was previously considered by Arveson and Josephson [W. Arveson, K. Josephson, Operator algebras and measure preserving automorphisms II, J. Funct. Anal. 4 (1969), 100–134.], Peters [J. Peters, Semicrossed products of C*-algebras, J. Funct. Anal. 59 (1984), 498–534.], Hadwin and Hoover [D. Hadwin, T. Hoover, Operator algebras and the conjugacy of transformations, J. Funct. Anal. 77 (1988), 112–122.] and Power [S. Power, Classification of analytic crossed product algebras, Bull. London Math. Soc. 24 (1992), 368–372.]. We also obtain a complete classification of all semicrossed products of the form , where denotes the disc algebra and 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.