The Szlenk index of C*-algebras

Lixin Cheng, Zhizheng Yu

Published 2023-06-14Version 1

Let $\mathcal A$ be a infinite dimensional C*-algebra. We show that the Szlenk index of $\mathcal A$ is $\Gamma'(i(\mathcal A))$, where $i(\mathcal A)$ is the noncommutative Cantor-Bendixson index, $\Gamma'(\xi)$ is the minimum ordinal number which is greater than $\xi$ of the form $\omega^\zeta$ for some $\zeta$ and we agree that $\Gamma'(\infty)=\infty$. As a application, we compute the Szlenk index of a C*-tensor product $\mathcal A\otimes_\beta\mathcal B$ of non-zero C*-algebras $\mathcal A$ and $\mathcal B$ in terms of $Sz(\mathcal A)$ and $Sz(\mathcal B)$. When $\mathcal A$ is a separable C*-algebra, we show that there has some $a\in \mathcal A_h$ such that $Sz(\mathcal A)=Sz(C^\ast(a))$, where $C^\ast(a)$ is the C*-subalgebra of $\mathcal A$ generated by $a$.