Approximately diagonalizing matrices over C(Y)

Abstract

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 (Equation Presented) 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.