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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Dynamical systems from function algebras

Authors: Tim Pennings and Justin Peters
Journal: Proc. Amer. Math. Soc. 105 (1989), 80-86
MSC: Primary 46J10; Secondary 43A45, 46L30, 46L55
MathSciNet review: 973840
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Abstract: Let $ X$ be compact Hausdorff, $ \Sigma $ the natural numbers or integers, $ \varphi :X \to X$, and $ \{ {\varphi ^k}:k \in \Sigma \} $ a (semi)group of continuous functions from $ X$ to $ X$. Given the dynamical system $ (X,\varphi ,\Sigma )$, let $ \mathfrak{A}$ be a $ \Sigma $-invariant $ {C^*}$-algebra of bounded functions containing $ C(X)$. There is a natural extension $ (\hat X,\hat \varphi ,\Sigma )$ of $ (X,\varphi ,\Sigma )$ where $ \hat X$ is the spectrum of $ \mathfrak{A}$ and $ \hat \varphi $ is given by $ \hat \varphi (\hat x)f = \hat x(f \circ \varphi )$. If $ \mathfrak{A}$ has a dense subset of functions continuous on a cofinite set, then $ (\hat X,\hat \varphi ,\Sigma )$ inherits the properties of minimality and topological transitivity from $ (X,\varphi ,\Sigma )$ if $ \mathfrak{A}$ contains no point characteristic functions.

References [Enhancements On Off] (What's this?)

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Keywords: Dynamical system, extension, Gelfand transform, minimal, orbit, spectrum, topologically transitive
Article copyright: © Copyright 1989 American Mathematical Society