Strong topological transitivity and \(C^*\)-dynamical systems
Let \(\tau\) be an action of a group H on a \(C^*\)-algebra \({\mathcal A}\). The action is said to be strongly topological transitive if for each pair \(x_ i,y_ i\), \(i=1,...,n\) of finite sequences in \({\mathcal A}\) such that \(\sum^{n}_{i=1}x_ i\circ y_ i\) there exists an \(h\in H\) such that \(\sum^{n}_{i=1}x_ i\tau_ h(y_ i)\neq 0\). If \({\mathcal A}\) is abelian, this reduces to the usual notion of topological transitivity for dynamical systems. The main result of this paper is a version of Tannaka's duality theorem: Let (\({\mathcal A},H,\tau)\) be a topologically transitive \(C^*\)-dynamical system and let \(\alpha\) be an action of a compact group G on \({\mathcal A}\) commuting with \(\tau\). If \(\beta\) is an automorphism of \({\mathcal A}\) such that \([\beta,\tau]=0\) and \(\beta (x)=x\) for all x in the fixed point algebra for \(\alpha\), it follows that \(\beta =\alpha_ g\) for some \(g\in G.\) The notion of strong topological transitivity is compared with the weaker notion of topological transitivity introduced in [\textit{R. Longo} and \textit{C. Peligrad}, J. Funct. Anal. 58, 157-174 (1984; Zbl 0556.46039)] and the two notions are shown to be equivalent in several cases. This question has recently been clarified further in [\textit{O. Bratteli, D. E. Evans} and \textit{A. Kishimoto}, Strong topological transitivity of \(C^*\)- dynamical systems II, preprint (1986)], but it is still an open problem in general whether the two notions are equivalent or not.
topologically transitive \(C^*\)-dynamical system
scientific article; zbMATH DE number 3953587