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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Cuntz, Joachim / Huybrechts, Daniel / Hwang, Jun-Muk

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Isomorphisms between topological conjugacy algebras

Kenneth R. Davidson1 / Elias G. Katsoulis2

1Pure Mathematics Department, University of Waterloo, Waterloo, ON N2L–3G1, Canada. e-mail: krdavids@uwaterloo.ca

2Department of Mathematics, East Carolina University, Greenville, NC 27858, USA. e-mail: KatsoulisE@mail.ecu.edu

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2008, Issue 621, Pages 29–51, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/CRELLE.2008.057, July 2008

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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.

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