Approximately diagonalizing matrices over C(Y)

2009/09/09 by Huaxin Lin, Lin, Huaxin
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #L4635 #Operator Algebras (math.OA) #math.FA #math.OA #msc:L4635

paper · pdf · doi:10.48550/arxiv.0909.1598

arxiv created 2009/09/09 · openalex publication_date 2009/09/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \rm dimY≤ 2. It is proved that there exists a sequence of n continuous maps \alfa_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 lim_m→∞ ∑_i=1ⁿ f(\alfa_i,m)p_i,m=ϕ(f) for all f∈ 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 \rm dimY≥ 3.

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