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 $Sz(\mathcal{A})={\Gamma }^{\prime }(i(\mathcal{A}))$ and $Dz(\mathcal{A})=Sz(L_p(\mathcal{A}))=\omega Sz(\mathcal{A})=\omega {\Gamma }^{\prime }(i(\mathcal{A}))$, where $i(\mathcal{A})$ is the noncommutative Cantor-Bendixson index, ${\Gamma }^{\prime }(\xi )$ is the minimum ordinal number which is greater than $\xi$ of the form ${\omega }^{\zeta }$ for some $\zeta$ and we agree that ${\Gamma }^{\prime }(\infty )=\infty$ 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 $a$.