Embeddings of von Neumann algebras into uniform Roe algebras and quasi-local algebras

TL;DR

This paper investigates which von Neumann algebras can be embedded into uniform Roe and quasi-local algebras, showing that only certain finite-dimensional matrix algebra products embed, with some restrictions on their structure.

Contribution

It characterizes the von Neumann algebras embeddable into these algebras, proving that only products of matrix algebras with bounded sizes can embed, and excludes embeddings of $L_[0,1]$.

Findings

Only products of matrix algebras with bounded sizes embed.

$L_[0,1]$ does not embed into these algebras.

Embedded von Neumann algebras are essentially finite-dimensional.

Abstract

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.