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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