Approximate diagonalization of self--adjoint matrices over $C(M)$

TL;DR

This paper characterizes when self-adjoint matrices over continuous functions on a compact space are approximately diagonalizable, linking the property to the space's topological dimension and cohomology groups.

Contribution

It establishes necessary and sufficient conditions for approximate diagonalization of matrices over $C(M)$ based on topological and cohomological properties of $M$.

Findings

Approximate diagonalization of self-adjoint matrices over $C(M)$ occurs iff $ ext{dim } M extless= 2$ and $ ext{H}^2(M, b Z) ext{ is trivial}$.

Unitary matrices over $C(M)$ are approximately diagonalizable iff $ ext{dim } M extless= 2$, with trivial first and second cohomology groups.

The results connect operator algebra properties with topological invariants of the underlying space.

Abstract

Let $M$ be a compact Hausdorff space. We prove that in this paper, every self--adjoint matrix over $C(M)$ is approximately diagonalizable iff $\dim M\le 2$ and $\HO^2(M,\mathbb Z)\cong 0$. Using this result, we show that every unitary matrix over $C(M)$ is approximately diagonalizable iff $\dim M\le 2$, $\HO^1(M,\mathbb Z)\cong\HO^2(M,\mathbb Z)\cong 0$ when $M$ is a compact metric space.