1-Bounded Entropy for $C^{\ast }$-Algebras

Abstract

Towards a question of Voiculescu, two notions of $1$-bounded entropy, $h$ and $h^{\text{top}}$, are defined for $C^{\ast }$-algebras. The quantity $h$ involves the tracial completion with respect to all traces while the quantity $h^{\text{top}}$ depends on operator norm microstates. It is demonstrated that $h^{\text{top}}(\mathcal{A})$ does not exceed $h(\mathcal{A})$ for all $C^{\ast }$-algebras $\mathcal{A}$. Moreover, a variational principle enabling the computation of $h$ is introduced. The quantity $h$ is computed in various examples, and the computation is used to justify structural conclusions such as $C^{\ast }$-primeness, crossed product indecomposability, and free indecomposability. These notions of entropy are generalized to operator systems, where it is demonstrated that both may be computed on a basis.

35 pages, 1 figure