Approximately diagonalizing matrices over C(Y)

Abstract

Let $X$ be a compact metric space which is locally absolutely retract and let $ϕ: C(X)\to C(Y, M_n)$ 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 $$ \lim_{m\to\infty} \sum_{i=1}^n f(\alfa_{i,m})p_{i,m}=ϕ(f) for all f\in C(X). $$ This is closely related to the Kadison diagonal matrix question. It is also shown that this approximate diagonalization could not hold in general when ${\rm dim}Y\ge 3.$