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Approximately diagonalizing matrices over \(C(Y)\). (English) Zbl 1262.46039
Summary: Let \(X\) be a compact metric space which is a locally absolute retract and let \(\varphi: C(X) \to C(Y,M_n)\) be a unital homomorphism, where \(Y\) is a compact metric space with \(\dim Y \leq 2\). It is proved that there exists a sequence of \(n\) continuous maps \(a_{i,m}:Y\to X\) (\(i=1,2,\dots,n\)) and a sequence of sets of mutually orthogonal rank-one projections \(\{p_{1,m},p_{2,m},\dots,p_{n,m}\} \subset C(Y,M_n)\) such that \[ \lim_{m\to\infty} \sum_{i=1}^n f(a_{i,m})p_{i,m} = \varphi(f) \quad \text{for all } f\in C(X). \] This is closely related to the Kadison diagonal matrix question. It is also shown that this approximate diagonalization could not hold in general when \(\dim Y\geq 3\).

MSC:
46L05 General theory of \(C^*\)-algebras
54E45 Compact (locally compact) metric spaces
54F45 Dimension theory in general topology
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