矩阵代数嵌入到万有Roe代数中
Embeddings of matrix algebras into the universal Roe algebra
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中文总结 AI 辅助
本文研究万有Roe代数的嵌入性质，证明其可嵌入$\prod_n\mathbb{M}_n$，否定Ozawa定理的推广，并得到置换算子凸包刻画及可分拟对角代数的嵌入结果。
中文摘要 AI 辅助
V. Manuilov引入了万有Roe代数，即与自然数集$\mathbb{N}$上最大的一致局部有限粗结构$\mathrm{Univ}$相关联的一致Roe代数，并引入了若干变体。我们通过$\mathrm{C}^*$-代数的可嵌入性来研究这些代数。在众多结果中，我们证明了万有Roe代数承认从$\prod_n\mathbb{M}_n$到其中的嵌入，从而回答了Farah的一个问题。因此，Ozawa关于$\prod_n\mathbb{M}_n$在一致局部有限度量空间的一致Roe代数中的不可嵌入定理并不推广到任意一致局部有限粗空间。然而，可嵌入性区分了Manuilov引入的变体以及$\mathrm{Univ}$的拟局部代数，但不区分乘子代数$M(\mathbb{I})$。作为副产品，我们还获得了Birkhoff问题111的算子范数类比：置换算子的范数闭凸包恰好是双重次随机算子集合。此外，我们证明了每个可分$\mathrm{C}^*$-代数都嵌入到$\mathrm{Univ}$的拟局部代数中。尽管我们不知道类似陈述是否对万有Roe代数成立，但我们证明了每个可分拟对角$\mathrm{C}^*$-代数都嵌入其中，从而嵌入到某个étale局部紧第二可数群胚的约化$\mathrm{C}^*$-代数中。
英文摘要
V. Manuilov introduced the universal Roe algebra, namely the uniform Roe algebra associated with the largest uniformly locally finite coarse structure $\mathrm{Univ}$ on $\mathbb{N}$, together with several variants. We study these algebras through embeddability properties of $\mathrm{C}^*$-algebras. Among other results, we show that the universal Roe algebra admits an embedding from $\prod_n\mathbb{M}_n$, answering a question by Farah. Thus, Ozawa's non-embeddability theorem for $\prod_n\mathbb{M}_n$ in uniform Roe algebras of uniformly locally finite metric spaces does not extend to arbitrary uniformly locally finite coarse spaces. Nevertheless, embeddability properties distinguish the variants introduced by Manuilov and the quasi-local algebra of $\mathrm{Univ}$, apart from the multiplier algebra $M(\mathbb{I})$. As a by-product, we also obtain an operator-norm analogue of Birkhoff's Problem 111: the norm-closed convex hull of permutation operators is exactly the set of doubly substochastic operators. Moreover, we show that every separable $\mathrm{C}^*$-algebra embeds into the quasi-local algebra of $\mathrm{Univ}$. Although we do not know whether the analogous statement holds for the universal Roe algebra, we prove that every separable quasidiagonal $\mathrm{C}^*$-algebra embeds into it, and hence into the reduced $\mathrm{C}^*$-algebra of some étale locally compact second countable groupoid.
发表机构
- RIMS, Kyoto University（京都大学数理解析研究所）
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