Isomorphism invariants for multivariable C*-dynamics

Abstract

To a given multivariable C*-dynamical system $(A,\alpha )$ consisting of *-automorphisms, we associate a family of operator algebras $\mathrm{alg}(A,\alpha )$, which includes as specific examples the tensor algebra and the semicrossed product. It is shown that if two such operator algebras $\mathrm{alg}(A,\alpha )$ and $\mathrm{alg}(B,\beta )$ are isometrically isomorphic, then the induced dynamical systems $(\hat{A},\hat{\alpha })$ and $(\hat{B},\hat{\beta })$ 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 $\mathrm{alg}(A,\alpha )$ and $\mathrm{alg}(B,\beta )$ are isometrically isomorphic, then the associated correspondences $X_{(A,\alpha )}$ and $X_{(B,\beta )}$ 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.