Uniform version of Weyl-von Neumann theorem

TL;DR

This paper provides a quantified version of the Weyl-von Neumann theorem, enabling uniform approximations of compact operators and simplifying the representation of uniform K-homology classes on a fixed Hilbert space.

Contribution

It introduces a quantified approximation approach to the Weyl-von Neumann theorem, facilitating uniform K-homology representations for broad classes of spaces.

Findings

Estimates ranks of approximants in Voiculescu's theorem

Simplifies uniform K-homology representations

Shows all classes can be represented on a fixed Hilbert space

Abstract

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.