Uniform version of Weyl-von Neumann theorem

We prove a "quantified" version of the Weyl-von Neumann theorem, more precisely, we estimate the ranks of approximants to compact operators appearing in the Voiculescu's theorem applied to commutative algebras. This allows considerable simplifications in uniform K-homology theory, namely it shows that one can represent all the uniform K-homology classes on a fixed Hilbert space with a fixed *-representation of C_0(X), for a large class of spaces X

Non-K-exact uniform Roe C*-algebras

We prove that uniform Roe C*-algebras C*uX associated to some expander graphs X coming from discrete groups with property (τ) are not K-exact. In particular, we show that this is the case for the expander obtained as Cayley graphs of a sequence of alternating groups (with appropriately chosen generating sets)

Uniform local amenability

The main results of this paper show that various coarse (‘large scale’) geometric properties are closely related. In particular, we show that prop- erty A implies the operator norm localisation property, and thus that norms of operators associated to a very large class of metric spaces can be effectively estimated.The main tool is a new property called uniform local amenability. This property is easy to negate, which we use to study some ‘bad’ spaces. We also generalise and reprove a theorem of Nowak relating amenability and asymptotic dimension in the quantitative setting

Quasi-locality and Property A

Let X be a metric space with bounded geometry, , and let E be a Banach space. The main result of this paper is that either if X has Yu's Property A and , or without any condition on X when , then quasi-local operators on belong to (the appropriate variant of) the Roe algebra of X. This generalises the existing results of this type by Lange and Rabinovich, Engel, Tikuisis and the first author, and Li, Wang and the second author. As consequences, we obtain that uniform -Roe algebras (of spaces with Property A) are closed under taking inverses, and another condition characterising Property A, akin to the operator norm localisation for quasi-local operators