Authors: Baudier, Florent P.
Braga, Bruno M.
Farah, Ilijas 
Vignati, Alessandro
Willett, Rufus
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Embeddings of von Neumann algebras into uniform Roe algebras and quasi-local algebras
Journal: Journal of Functional Analysis
Volume: 286
Issue: 1
First page: 110186
Issue Date: 2024
Rank: ~M21
ISSN: 0022-1236
DOI: 10.1016/j.jfa.2023.110186
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 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.
Keywords: Coarse geometry | Uniform Roe algebras | Von Neumann algebras
Publisher: Elsevier

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