Lin, Huaxin Approximately diagonalizing matrices over \(C(Y)\). (English) Zbl 1262.46039 Proc. Natl. Acad. Sci. USA 109, No. 8, 2842-2847 (2012). 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 \(\dim Y\geq 3\). Cited in 5 Documents MSC: 46L05 General theory of \(C^*\)-algebras 54E45 Compact (locally compact) metric spaces 54F45 Dimension theory in general topology Keywords:Kadison diagonal matrix question; approximate diagonalization PDFBibTeX XMLCite \textit{H. Lin}, Proc. Natl. Acad. Sci. USA 109, No. 8, 2842--2847 (2012; Zbl 1262.46039) Full Text: DOI arXiv References: [1] BULL AMER MATH SOC NS 8 pp 84– (1983) · Zbl 0518.46047 · doi:10.1090/S0273-0979-1983-15091-7 [2] AMER J MATH 106 pp 1451– (1984) · Zbl 0585.46048 · doi:10.2307/2374400 [3] J FUNCT ANAL 59 pp 65– (1984) · Zbl 0554.46026 · doi:10.1016/0022-1236(84)90053-3 [4] INDAG MATH 32 pp 96– (1970) [5] J FUNCT ANAL 160 pp 466– (1998) · Zbl 0939.46033 · doi:10.1006/jfan.1998.3261 [6] MEM AMER MATH SOC 205 pp 1– (2010) [7] J OPERATOR THEORY 15 pp 15– (1986) [8] ANN OF MATH 104 pp 585– (1976) · Zbl 0361.46067 · doi:10.2307/1970968 [9] FIELDS INST COMMUN WATERLOO ON 13 pp 193– (1997) [10] J OPERATOR TH 40 pp 217– (1998) [11] AMER J MATH 72 pp 214– (1950) · Zbl 0034.29601 · doi:10.2307/2372148 [12] J FUNCT ANAL 258 pp 1822– (2010) · Zbl 1203.46038 · doi:10.1016/j.jfa.2009.11.020 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.