Approximately diagonalizing matrices over C(Y)

Let X be a compact metric space which is locally absolutely retract and let ϕ: C(X) → C(Y,M(n)) be a unital homomorphism, where Y is a compact metric space with dim Y ≤ 2. It is proved that there exists a sequence of n continuous maps α(i,m): Y → 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)} C(Y,M(n)) such that [see formula]. This is closely related to the Kadison diagonal matrix question. It is also shown that this approximate diagonalization could not hold in general when dim Y ≥ 3.