Szlenk and $w^\ast$-dentability indices of C$^\ast$-algebras

Abstract

Let $\mathcal A$ be a infinite dimensional C*-algebra and $1<p<\infty$. We compute the Szlenk index of $\mathcal A$ and $L_p(\mathcal A)$, and show that $Sz(\mathcal A)=\Gamma'(i(\mathcal A))$ and $Dz(\mathcal A)=Sz(L_p(\mathcal A))=\omega Sz(\mathcal A)=\omega\Gamma'(i(\mathcal A))$, 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 $\Gamma'(\infty)=\infty$ and $\omega\cdot\infty=\infty$. As a application, we compute the Szlenk index [respectively, $w^\ast$-dentability 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 $Sz(\mathcal B)$ [respectively, $Dz(\mathcal A)$ and $Dz(\mathcal B)$]. When $\mathcal A$ is a separable C*-algebra, we show that there exists $a\in \mathcal A_h$ such that $Sz(\mathcal A)=Sz(C^\ast(a))$ and $Dz(\mathcal A)=Dz(C^\ast(a))$, where $C^\ast(a)$ is the C*-subalgebra of $\mathcal A$ generated by $a$.