Properties of topological dynamical systems and corresponding \(C^*\)- algebras
By a topological dynamical system \(\Sigma =(X,G)\) is understood a group action of a discrete group G on a compact Hausdorff space X. Let C(X) denote the algebra of continuous functions on X and \(X^ t\) the set of fixed points of the action \(x\to xt\) for \(t\in G\). The authors study the transformation group \(C^*\)-algebra A(\(\Sigma\)) associated with \(\Sigma =(X,G)\). They show that if each non-zero ideal in A(\(\Sigma\)) has an intersection with the subalgebra C(X) of A(\(\Sigma\)), then the interior of \(X^ t\) is empty for all \(t\neq \ell\), and the converse implication holds if the group G is amenable. This means that A(\(\Sigma\)) is simple if and only if the action of G on X is minimal, Int \(X^ t=\emptyset\) for all \(t\neq \ell\) and G is amenable.
scientific article; zbMATH DE number 4194693