Embeddings of Uniform Roe Algebras

Abstract

In this paper, we study embeddings of uniform Roe algebras. Generally speaking, given metric spaces X and Y , we are interested in which large scale geometric properties are stable under embedding of the uniform Roe algebra of X into the uniform Roe algebra of Y . Contents 1. Introduction 1 2. Preliminaries 5 3. Coarse-like maps and quasi-locality 8 4. Making embeddings strongly continuous 13 5. Embeddings and geometry preservation 16 6. Embeddings onto hereditary subalgebras 22 7. Digression on our geometric property 27 8. Open problems 30 References 32

On the uniform Roe algebra as a Banach algebra and embeddings of ℓp uniform Roe algebras

“…We would like to emphasize that all the results below holds for p = 2 and are new even for p = 2. The following should be compared with [4 Our results hold in a Banach algebraic setting, and we do not require our maps to be isometric. We can therefore forget the structure of * -algebra when dealing with uniform Roe C *algebras, and also eliminate the condition requiring isometric isomorphisms in [ When p = 2, we have the next corollary.…”

“…In this paper, we study the generalization of uniform Roe ‐algebras to algebras of operators on , for , and their mutual embeddings, by generalizing to the case the results obtained for uniform Roe ‐algebras in [4]. Since we only treat such objects from a Banach algebraic point of view, as a consequence we are able to remove certain hypotheses from results of Farah and the authors [2, 4], Špakula and Willett [17], and Chung and Li [6].…”

“…Our goal is to study mutual embeddings of uniform Roe algebras, in parallel with the work in the ‐setting carried on in [4]. Different from the isometric isomorphism case studied in [6], the geometric assumption present in the ‐case cannot be removed, as there are partial isometries on which are not merely given by a permutation of together with a change of signs.…”

“…We would like to emphasize that all the results below holds for and are new even for . The following should be compared with [4, Theorem 1.2]. Theorem Let , and and be uniformly locally finite (u.l.f.)…”

“…Also, note that one cannot hope to obtain in Theorem 1.1 that coarsely embeds into . Indeed, it is easy to see that embeds into by a compact preserving map, but does not coarsely embed into (see [4, Proposition 2.4]). However the conclusion of Theorem 1.1 is enough to guarantee that if has finite asymptotic dimension, finite decomposition complexity, or property A, then has the same respective property [4, Proposition 2.5, Corollary 5.8 and Proof of Corollary 1.3(ii)].…”