论文标题
von Neumann代数的嵌入到均匀的ROE代数和准局部代数中
Embeddings of von Neumann algebras into uniform Roe algebras and quasi-local algebras
论文作者
论文摘要
我们研究了哪些von Neumann代数可以嵌入均匀的ROE代数和与统一的本地有限度量空间$ x $相关的准局部代数。在虚弱的假设下，这些$ \ mathrm {c}^*$ - 代理包含$ \ prod_ {k} \ mathrm {m} _ {n_k}（\ mathbb c）$的嵌入式拷贝我们的目的是表明它们不能包含任何其他冯·诺伊曼代数。 我们的主要结果之一表明，即使是由于不必要的$*$ - 同构的，即使是通过不必要的正常$*$ - 也不会嵌入这些代数中的任何一个。特别是，它遵循von Neumann代数的结构理论，即嵌入此类代数的任何von Neumann代数都必须形式为$ \ prod_ {k} \ mathrm {m} _ {m} _ {n_k} _ {n_k}（\ mathbb c）$ for Ligalable（可能有限的（可能是有限的），在其他假设下，我们还表明，序列$（n_k）_k $必须有限：换句话说，唯一嵌入的von neumann代数是``显而易见''。
We study which von Neumann algebras can be embedded into uniform Roe algebras and quasi-local algebras associated to a uniformly locally finite metric space $X$. Under weak assumptions, these $\mathrm{C}^*$-algebras contain embedded copies of $\prod_{k}\mathrm{M}_{n_k}(\mathbb C)$ for any \emph{bounded} countable (possibly finite) collection $(n_k)_k$ of natural numbers; we aim to show that they cannot contain any other von Neumann algebras. One of our main results shows that $L_\infty[0,1]$ does not embed into any of those algebras, even by a not-necessarily-normal $*$-homomorphism. In particular, it follows from the structure theory of von Neumann algebras that any von Neumann algebra which embeds into such algebra must be of the form $\prod_{k}\mathrm{M}_{n_k}(\mathbb C)$ for some countable (possibly finite) collection $(n_k)_k$ of natural numbers. Under additional assumptions, we also show that the sequence $(n_k)_k$ has to be bounded: in other words, the only embedded von Neumann algebras are the ``obvious'' ones.
