Isomorphism invariants for multivariable \(C^\ast\)-dynamics
Let \((A, \alpha)\) be a multivariable \(C^*\)-dynamical system. A family of operator algebras \(\mathrm{alg}(A, \alpha)\) is associated to \((A, \alpha)\) which includes the tensor algebra and the semicrossed product. In this paper, isomorphism invariants for operator algebras associated with multivariable \(C^*\)-dynamical systems are studied. If \((A, \alpha)\) is an automorphic multivariable \(C^*\)-dynamical system, then it induces an homeomorphic multivariable system \((\hat{A}, \hat{\alpha})\) on the spectrum \(\hat{A}\) of \(A\). Let \((A, \alpha)\) and \((B, \beta)\) be automorphic multivariable \(C^*\)-dynamical systems. The main result of the paper is that, if there exist associated operator algebras \(\mathrm{alg}(A, \alpha)\) and \(\mathrm{alg}(B, \beta)\) which are isometrically isomorphic, then the induced multivariable systems \((\hat{A}, \hat{\alpha})\) and \((\hat{B}, \hat{\beta})\) are piecewise conjugate. Several other results of independent interest are also obtained.
scientific article; zbMATH DE number 6368719
scientific article