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

Abstract

We work on p uniform Roe algebras associated to metric spaces, and on their mutual embedding. We generalize results of Farah and the authors to mutual embeddings of uniform Roe algebras of operators on p spaces. Simultaneously, we obtain rigidity results for the classic uniform Roe C *-algebras which depend only on their Banach algebra structure.

General Uniform Roe algebra rigidity

“…-We have proved that the closing-off arguments used in the proof of Proposition 3.2 were not needed. This is because by Corollary 3.3 (7) all relevant properties reflect to all countably generated reducts.…”

“…By Proposition 3.2 (3), ( 7) implies (3). On the other hand, the property "all ghost operators in C * u (X, E) are compact" is preserved by taking reducts, by Proposition 2.6, hence (3) implies (7), and so the two are equivalent.…”

“…(6) Every countably generated reduct of (X, E) has ONL. (7) For every countably generated reduct (X, F) of (X, E), all ghost operators in C * u (X, F) are compact. (8) For every countably generated reduct (X, F) of (X, E), C * u (X, F) is nuclear.…”

“…All the known proofs of (strong or weak) rigidity, starting with [26], proceed by analysis of the sets Y x,δ , along the following three steps. (7) (1) One finds δ > 0 such that Y x,δ and X y,δ are nonempty for all x ∈ X and y ∈ Y . (2) One proves that the sets x∈X Y x,δ × Y x,δ and y∈Y X y,δ × X y,δ are entourages (in Y and X respectively) for all δ > 0.…”

“…(4) If y and y are distinct elements of Y , then X y,η and X y ,η are disjoint. (7) A similar approach is used when dealing with strong or weak rigidity for embeddings from the existence of C * -embeddings onto hereditary subalgebras.…”