C*-algebras of Irrational Time Homeomorphisms of Suspensions

The crossed product construction provides a bridge §between operator algebras and dynamical systems. For §instance, it is known that two ergodic non singular §dynamical systems are orbit equivalent if and only §if their associated von Neumann crossed products are §isomorphic. In the topological setting, it is known §that two minimal dynamical systems on Cantor spaces §are strong orbit equivalent if and only if their §associated C crossed products are isomorphic. In §the literature there are examples of dynamical §systems which are not orbit equivalent yet their §associated C crossed products are either isomorphic §or have isomorphic Elliott invariant. These §examples, however, deal with spaces which are either §disconnected or have dimension greater than one. By §considering irrational time homeomorphisms of §suspensions, this book provides examples of §dynamical systems on 1-dimensional compact connected §metric spaces with the required property. The §analysis in the book suggests that conditions to §extend the given result about Cantor minimal systems §might need to take into account the topological §entropy of the systems involved.