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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 [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.

SUBMITTER: Lin H

PROVIDER: S-EPMC3286972 | biostudies-literature | 2012 Feb

REPOSITORIES: biostudies-literature

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Approximately diagonalizing matrices over C(Y).

Lin Huaxin H

Proceedings of the National Academy of Sciences of the United States of America 20120208 8

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 diagon ...[more]

PMID: 22323593

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