On the uniform Roe algebra as a Banach algebra and embeddings of $\ell_p$ uniform Roe algebras

Abstract

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

On Banach algebras of band-dominated operators and their order structure

“…such that each A i is the graph of a partial bijection 8 of N. Moreover, the partion can be taken so that d(n 1 , n 2 ) > r and d(m 1 , m 2 ) > r for all i ∈ {1, . .…”

“…As mentioned in the introduction, if (X, d) is a u.l.f. metric space and E is the standard basis of ℓ p , the uniform Roe algebra B u (d, E) is frequently denoted by B p u (X) (see [8,10,31]). In this case, B p u (X) contains the compact operators if and only if Proof.…”

“…Inspired by C * -algebra theory, we say that B is a hereditary subalgebra of A if BAB ⊂ B. The following was proved in [8,Lemma 3.11] for E being the standard basis of ℓ p . Since the exactly same proof holds for any basis, we omit its proof here.…”

“…The following follows straightforwardly from the definition of coarse-like maps (cf. [8,Lemma 3.2]). Theorem 7.9.…”

“…for all n, m ∈ N (see [8,Claim 3.13] for similar computation). Therefore, since our choice of f and g imply that e g(n),g(n) Φ(e n,n ) ≥ δ and Φ(e n,n )e f (n),f (n) ≥ δ, we have that e g(n),g(n) Φ(e m,n )e f (m),f (m) > θ 2 δ 2 for all n, m ∈ N.…”