Isomorphism Invariants for Multivariable C*-Dynamics

TL;DR

This paper establishes that isometric isomorphisms between operator algebras from multivariable C*-dynamical systems imply piecewise conjugacy of their spectra, providing new invariants and connections between algebraic and dynamical properties.

Contribution

It introduces a new invariant for multivariable C*-dynamical systems based on isometric isomorphisms of associated operator algebras, linking algebraic isomorphisms to dynamical conjugacy.

Findings

Isometric isomorphism implies piecewise conjugacy of spectra.

Associated correspondences are unitarily equivalent under algebra isomorphism.

Semicrossed product isomorphism implies tensor algebra isomorphism.

Abstract

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