Approximately diagonalizing matrices over C(Y).
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- Record sourced from PubMed, PMID 22323593.
- Also identified by DOI 10.1073/pnas.1101079108 and PMC identifier 3286972.
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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 [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.