On extensions of dynamical systems by function algebras
by
Domingo Alcaraz
Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena
Coauthors: Manuel Sanchis (Departament de Matemàtiques, Universitat Jaume I de Castelló (Spain))

Following Penning and Peters (Proc. Amer. Math. Soc., 1989) an extension ([^X], \Sigma) of a flow (X, \Sigma) generated by a continuous function \phi:X --> X ( where \Sigma = Z if \phi is a homeomorphism, and \Sigma = N otherwise) is called a tame extension of (X, \Sigma) if (i) and (ii) are satisfied: (i) [^X] is the spectrum of a \phi-invariant C\sp *-algebra U of functions, C(X) subset U subset B(X), where C(X) and B(X) are the continuous and bounded C-valued functions on X, respectively, and p:[^X] --> X maps [^x] to [^x]|\sbC(X); (ii) there is a \Sigma-invariant subset E subset U, which generates U, consists of functions with at most a finite number of points of discontinuity, and contains no delta functions \delta\sb x.

The main result in the paper cited above states that for every x\sb 0 in X and every [^x]\sb 0 in p\sp -1(x\sb 0), the orbit of [^x]\sb 0 is dense whenever the orbit of x\sb 0 is dense. As a direct corollary, it follows that topological transitivity [resp. minimality] passes from (X, j, \Sigma) to ([^X], [^(j)], \Sigma).

In this talk we improve Pennings-Peters' result to a wide class of C\sp *-algebras, namely to C\sp *-algebras U such that U is topologically generated by a subset of functions each of which is continuous off a dispersed set, and has no characteristic functions of singletons.

Date received: February 28, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafw-48.