Approximately diagonalizing matrices over C(Y)

TL;DR

This paper demonstrates an approximate diagonalization of certain matrix-valued functions over compact spaces with dimension at most 2, extending the understanding of Kadison's diagonalization question, but shows limitations for higher dimensions.

Contribution

It establishes conditions under which matrices over C(Y) can be approximately diagonalized when dim Y ≤ 2, and highlights the failure of this in higher dimensions.

Findings

Approximate diagonalization is possible for dim Y ≤ 2.

Such diagonalization fails when dim Y ≥ 3.

The work relates to Kadison's diagonal matrix problem.

Abstract

Let $X$ be a compact metric space which is locally absolutely retract and let $\phi: C(X)\to C(Y, M_n)$ be a unital homomorphism, where $Y$ is a compact metric space with ${\rm dim}Y\le 2.$ It is proved that there exists a sequence of $n$ continuous maps $\alfa_{i,m}: Y\to X$ ($i=1,2,...,n$) and a sequence of sets of mutually orthogonal rank one projections $\{p_{1, m}, p_{2,m},...,p_{n,m}\}\subset C(Y, M_n)$ such that $$ \lim_{m\to\infty} \sum_{i=1}^n f(\alfa_{i,m})p_{i,m}=\phi(f) for all f\in C(X). $$ This is closely related to the Kadison diagonal matrix question. It is also shown that this approximate diagonalization could not hold in general when ${\rm dim}Y\ge 3.$