Dynamical systems of type \((m,n)\) and their \(C^\ast\)-algebras (Q2873986)

For given \( n,m \in \mathbb{Z},\) the authors consider a dynamical systems in which the disjoint union of \( n \) copies of a topological space is homeomorphic to \(m\) copies of that same space. The universal such system is shown to arise naturally from the study of a \(C^\ast\)-algebra denoted by \(\mathcal{O}_{m,n}\), which in turn is obtained as a quotient of the well-known Leavitt \(C^\ast\)-algebra \(L_{m,n}\), the process transforms the generating set of partial isometries of \(L_{m,n}\) into a tame set. The authors take the advantage of the existing literature on \(C^\ast\)-algebras generated by tame sets of partial isometries to describe \(\mathcal{O}_{m,n}\) as the crossed product associated to the partial action \(\theta^{u}\) of the free group \(\mathbb{F}^{m+n}\) on a compact space \(\Omega^{u}\). Also the authors consider the question of the existence of finite-dimensional representations of \(\mathcal{O}_{m,n}\) and \(\mathcal{O}_{m,n}^{r}\), and a trivial argument proves that when \(n \neq m\) then neither \(\mathcal{O}_{m,n}\) nor \(\mathcal{O}_{m,n}^{r}\) admit finite-dimensional representations. The authors conclude by proving that \(\mathcal{O}_{m,n}^{r}\) admits no finite-dimensional representation for all \(m,n \geq 2\).