To a given multivariable C*-dynamical system consisting of *-automorphisms, we associate a family of operator algebras , which includes as specific examples the tensor algebra and the semicrossed product. It is shown that if two such operator algebras and are isometrically isomorphic, then the induced dynamical systems and on the Fell spectra are piecewise conjugate in the sense of Davidson and Katsoulis.
In the course of proving the above theorem we obtain several results of independent interest. If and are isometrically isomorphic, then the associated correspondences and are unitarily equivalent. In particular, the tensor algebras are isometrically isomorphic if and only if the associated correspondences are unitarily equivalent. Furthermore, isomorphism of semicrossed products implies isomorphism of the associated tensor algebras.
In the case of multivariable systems acting on C*-algebras with trivial center, unitary equivalence of the associated correspondences reduces to outer conjugacy of the systems. This provides a complete invariant for isometric isomorphisms between semicrossed products as well.
Cite this article
Evgenios T. A. Kakariadis, Elias G. Katsoulis, Isomorphism invariants for multivariable C*-dynamics. J. Noncommut. Geom. 8 (2014), no. 3, pp. 771–787DOI 10.4171/JNCG/170