Approximately diagonalizing matrices over C(Y)

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