Topological entropy in exact \(C^*\)-algebras
\textit{D. Voiculescu} defined a notion of topological entropy for automorphisms of unital nuclear \(C^*\)-algebras in [Commun. Math. Phys. 170, No. 2, 249-281 (1995; Zbl 0824.46079)]. Based on the known fact of existence of a faithful nuclear representation \(\pi\) for the exact \(C^*\)-algebra \(A\) by \textit{E. Kirchberg} [J. Funct. Anal. 129, No. 1, 35-63 (1995; Zbl 0912.46059)], the author successfully extends Voiculescu's approximation approach to the class of exact \(C^*\)-algebras by replacing the identity map of the algebra with a faithful representation. The author first defines the topological entropy \(ht_A(\alpha)\) of an automorphism \(\alpha\) of an arbitrary exact \(C^*\)-algebra \(A\) with respect to a faithful (possibly degenerate) \(*\)-representation \(\pi: A\to B(H)\) of \(A\). It is proved that this entropy is independent of the representation (hence it is an invariant of the \(C^*\)-dynamical system \((A,\alpha)\)), and also that its definition agrees with Voiculescu's one in the case of a nuclear \(C^*\)-algebra. Next a limit formula for the entropy of inductive limit automorphisms is shown as well. Here is the main theorem: ``If \(A\) is a unital exact \(C^*\)-algebra, \(\alpha: G\to\Aut(A)\) a group homomorphism (with \(G\) discrete and Abelian) taking \(g\mapsto \alpha_g\) and \(\lambda_g\in A\times_\alpha G\) the unitary implementing the automorphism \(\alpha_g\in \Aut(A)\), then \(ht_A(\alpha_g)= ht_{A\times_\alpha G}(\text{Ad}+\lambda_g)\) holds.'' On a technical basis, the proof is greatly due to the Arveson extension theorem for completely positive maps, cf. \textit{W. B. Arveson} [Acta Math. 123(1969), 141-224 (1970; Zbl 0194.15701)].
scientific article; zbMATH DE number 1337019