Dynamical systems from function algebras

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.