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

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 C⁎-algebras contain embedded copies of ∏kMnk(C) for any bounded countable (possibly finite) collection (nk)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∞[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 ∏kMnk(C) for some countable (possibly finite) collection (nk)k of natural numbers. Under additional assumptions, we also show that the sequence (nk)k has to be bounded: in other words, the only embedded von Neumann algebras are the “obvious” ones.