\(C^*\)-algebras associated with shift dynamical systems
The authors study the \(C^*\)-algebras generated by shift operators. Those algebras can be considered as \(C^*\)-crossed products \(C(X)\times_{\sigma}Z,\) where C(X) is the \(C^*\)-algebra of all continuous functions on a compact space X and \(\sigma\) is a topologically transitive homeomorphism on X. Let X be a dual subgroup of a countable subgroup of the unit circle \({\mathbb{T}}\) with discrete topology. Then X is a monothetic compact abelian group, that is, there exists a homomorphism \(\phi\) from the group Z of all integers onto a dense subgroup of X. Thus the homeomorphism \(\sigma:x\to x+\phi (1)\) (\(x\in X)\), is topologically transitive. In this case, those algebras have been completely classified by \textit{M. A. Rieffel} [Pac. J. Math. 93, 415-429 (1981; Zbl 0499.46039)] and \textit{N. Riedel} [Pac. J. Math. 101, 153-161 (1982; Zbl 0504.46040)]. The authors extend their result to the case of an arbitrary subgroup of \({\mathbb{T}}\) and discuss several examples. Furthermore they study the relation between considering \(C^*\)-algebras and compact operators.
\(C^*\)-algebras generated by shift operators
scientific article; zbMATH DE number 3872079