Decompositions of topological dynamical systems and their transformation group \(C^*\)-algebras (Q1210376)

Given a topological dynamical system \(\Sigma= (X,\sigma)\) where \(X\) is a compact metric space with a homeomorphism \(\sigma\), we sometimes meet the situation that the system \(\Sigma\) is decomposed into the disjoint union of subdynamical systems \(\{\Sigma_ \gamma= (X_ \gamma, \sigma\mid X_ \gamma)\mid \gamma\in \Gamma\}\) where each \(X_ \gamma\) is an invariant closed subset of \(X\). For instance, it is known that when \(\sigma\) is distal this is always the case in which \(\Sigma_ \gamma\) turns out to be minimal. Thus, in such a case the transformation group \(C^*\)-algebra \(A(\Sigma)\) associated to the system \(\Sigma\) may be considered as a connection of the family of transformation group \(C^*\)- algebras, \(\{ A(\Sigma_ \gamma)\mid \gamma\in\Gamma\}\) of those topological subsystems \(\Sigma_ \gamma\). It is the purpose of the present paper to show that, under what conditions, the algebra \(A(\Sigma)\) becomes the algebra of all continuous operator fields over the fibred space \(\{\Gamma\mid A(\Sigma_ \gamma)\}\) in a fairly general setting for topological dynamical systems. There are recent results closely related to our present discussions, namely those by \textit{D. P. Williams} [J. Aust. Math. Soc., Ser. A 47, No. 2, 226-235 (1989; Zbl 0687.46044)] and \textit{M. A. Rieffel} [Math. Ann. 283, No. 4, 631-643 (1989; Zbl 0664.46063)]. The main difference between their results and ours is the following: they start using the fibred space \(\{Y, A(t), {\mathcal F}\}\) of \({\mathcal C}^*\)-algebras with a compatible action of a group \(G\) on the algebra of continuous operator fields on \(\{Y, A(t), {\mathcal F}\}\) and ask whether the situation is still compatible or not in the fibered space \(\{Y, A(t)\times_{\alpha_ t} G\}\) of crossed product \({\mathcal C}^*\)-algebras \(A(t)\times_{\alpha_ t} G\)'s whereas we are interested in the conditions when such a starting situation arises from a given topological dynamical system. Throughout our arguments, a homomorphism between \(C^*\)-algebras always means a *- preserving homomorphism.