Szlenk and $w^\ast$-dentability indices of C$^\ast$-algebras
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866929302767927296 |
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| author | Cheng, Lixin Yu, Zhizheng |
| author_facet | Cheng, Lixin Yu, Zhizheng |
| contents | 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 $Sz(\mathcal A)=Γ'(i(\mathcal A))$ and $Dz(\mathcal A)=Sz(L_p(\mathcal A))=ωSz(\mathcal A)=ωΓ'(i(\mathcal A))$, where $i(\mathcal A)$ is the noncommutative Cantor-Bendixson index, $Γ'(ξ)$ is the minimum ordinal number which is greater than $ξ$ of the form $ω^ζ$ for some $ζ$ and we agree that $Γ'(\infty)=\infty$ and $ω\cdot\infty=\infty$. As a application, we compute the Szlenk index [respectively, $w^\ast$-dentability index] of a C*-tensor product $\mathcal A\otimes_β\mathcal B$ 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 $a$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_08515 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Szlenk and $w^\ast$-dentability indices of C$^\ast$-algebras Cheng, Lixin Yu, Zhizheng Functional Analysis Operator Algebras 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 $Sz(\mathcal A)=Γ'(i(\mathcal A))$ and $Dz(\mathcal A)=Sz(L_p(\mathcal A))=ωSz(\mathcal A)=ωΓ'(i(\mathcal A))$, where $i(\mathcal A)$ is the noncommutative Cantor-Bendixson index, $Γ'(ξ)$ is the minimum ordinal number which is greater than $ξ$ of the form $ω^ζ$ for some $ζ$ and we agree that $Γ'(\infty)=\infty$ and $ω\cdot\infty=\infty$. As a application, we compute the Szlenk index [respectively, $w^\ast$-dentability index] of a C*-tensor product $\mathcal A\otimes_β\mathcal B$ 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 $a$. |
| title | Szlenk and $w^\ast$-dentability indices of C$^\ast$-algebras |
| topic | Functional Analysis Operator Algebras |
| url | https://arxiv.org/abs/2306.08515 |