Conjugate dynamical systems on \(C^*\)-algebras (Q2878708)

The main results of the paper are as follows. If \((A,\alpha)\) and \((B,\beta)\) are unital \(C^*\)-algebra dynamical systems, where \(\alpha\) is either injective or surjective, then \(A \times_{\alpha} \mathbb{Z}_{+}\) and \(B \times_{\beta} \mathbb{Z}_{+}\) are isometrically isomorphic \(\Longleftrightarrow (A,\alpha), (B,\beta)\) are outer conjugate. Let \((A,\alpha), (B,\beta)\) be unital \(C^*\)-algebra dynamical systems. If one of some properties such as \(A\) has trivial center, \(A\) is abelian, \(A\) is finite or \(\alpha (A)'\) is finite, \(\alpha(R_{\alpha})=R_{\alpha}\), and \(\alpha(\operatorname{ann}(R_{\alpha})) \subseteq \operatorname{ann}(R_{\alpha})\) holds, then \(A \times_{\alpha} \mathbb{Z}_{+}\) and \(B \times_{\beta} \mathbb{Z}_{+}\) are isometrically isomorphic if and only if \((A,\alpha)\) and \((B,\beta)\) are outer conjugate.