Szlenk and w*-dentability indices of C*-algebras

ANALYSIS SEMINAR

Title: Szlenk and  ${\mathrm{<span\; style="font-family:"times\; new\; roman";">w</span>}}^{<span\; style="font-family:"times\; new\; roman";"> \ast </span>}$-dentability indices of C*-algebras 

Speaker：Zhizheng Yu (Harbin Institute of Technology) 

Time：2024-10-16（Wednesday），14:30-16:00

Venue：Minde Building B201-1 

Zoom ID：947 0981 8605，Password：477439

Abstract：

Let $A$ be an infinite dimensional C*-algebra and $1<p<\mathrm{ \infty }$. We compute the Szlenk index of $A$ and $L_p\left(A\right)$, and show that $Sz\left(A\right)= \Gamma \text{'}\left(i\left(A\right)\right)$ and $Dz\left(A\right)=Sz\left(L_p\left(A\right)\right)=\mathrm{ \omega }Sz\left(A\right)=\mathrm{ \omega } \Gamma \text{'}\left(i\left(A\right)\right)$, where $i\left(A\right)$ is the noncommutative Cantor-Bendixson index, $\Gamma \text{'}\left(\mathrm{ \xi }\right)$ is the minimum ordinal number which is greater than $\mathrm{ \xi }$ of the form ${\mathrm{ \omega }}^{\mathrm{ \zeta }}$ for some $\mathrm{ \zeta }$ and we agree that $\Gamma \text{'}\left(\mathrm{ \infty }\right)=\mathrm{ \infty }$ and $\mathrm{ \omega } \cdot \mathrm{ \infty }=\mathrm{ \infty }$. As an application, we compute the Szlenk index [respectively, $w^{ \ast }$-dentability index] of a C*-tensor product $A{ \otimes }_{\mathrm{ \beta }}B$ of non-zero C*-algebras $A$ and $B$ in terms of $Sz\left(A\right)$ and $Sz\left(B\right)$ [respectively, $Dz\left(A\right)$ and $Dz\left(B\right)$]. When $A$ is a separable C*-algebra, we show that there exists $a \in A_h$ such that $Sz\left(A\right)=Sz\left(C^{ \ast }\left(a\right)\right)$ and $Dz\left(A\right)=Dz\left(C^{ \ast }\left(a\right)\right)$, where $C^{ \ast }\left(a\right)$ is the C*-subalgebra of $A$ generated by $a$.

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