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Diagonalizing matrices over C(X). (English) Zbl 0554.46026

If A(x) is a normal \(n\times n\)-matrix, depending continuously on a parameter x in some compact Hausdorff space X, one may ask for the existence of a continuous unitary matrix function U(x), \(x\in X\), such that \(U(x)A(x)U(x)^*\) is diagonal for every x in X. If A(x) is multiplicity-free for each x, we show that A is diagonalizable when X is a 2-connected CW-complex. When multiplicity occur, the situation is much more complicated: A necessary condition for diagonalizability is (i) that the space X be sub-Stonean, i.e. every two disjoint, open \(\sigma\)- compact subsets of X have disjoint closures. An early reference to this obstruction is: L. Gillman and M. Henriksen [Trans. Am. Math. Soc. 82, 362-365 (1956; Zbl 0073.022)]. A survey of sub-Stonean spaces can be found in: K. Grove and G. K. Pedersen [J. Funct. Anal. 56, 124-143 (1984; Zbl 0539.54029)]. Other obstructions for diagonalizability arise if some closed subset \(X_ 0\) of X carries a non-trivial G-bundle, when G is a symmetric group or the circle group. The resulting necessary conditions are (ii) that the covering dimension for X be \(\leq 2\), (iii) that \(H^ 1(x_ 0,S_ m)=0\) for every \(X_ 0\subset X\) and all m; and (iv) that \(H^ 2(X_ 0,{\mathbb{Z}})=0\) for every \(X_ 0\subset X.\)
We show that on a space X, satisfying the conditions, (i)-(iv), diagonalization is always possible. As an example of such a space we have \(X=\beta (Y)\setminus Y\), for some infinite, contractible graph Y with Stone-Čech compactification \(\beta\) (Y). A simpler example (with a much simpler proof) occurs if X is a Stonean space (or just a closed subset of a Rickart space). Here we can even triangularize an arbitrary (continuous) matrix function A.
This paper is a reply to a question raised by R. V. Kadison [Bull. Am. Math. Soc. 8, 84-86 (1983; Zbl 0518.46047)] who shows that if \({\mathcal M}\) is an arbitrary von Neumann algebra and \({\mathfrak A}\) is a commutative \({}^*\)-subalgebra of \({\mathcal M}\otimes {\mathbb{M}}_ n\)- the \(n\times n\)- matrices over \({\mathcal M}\)- then there is a unitary U in \({\mathcal M}\otimes {\mathbb{M}}_ n\) such that U\({\mathfrak A}U^*\) is a diagonal algebra. In this setting our theorem characterizes those commutative \(C^*\)-algebras C(X), for which Kadison’s result is valid for a countably generated, commutative \({}^*\)-subalgebra \({\mathfrak A}\) of \(C(X)\otimes {\mathbb{M}}_ n\). If one insists on the diagonalizability of every commutative \({}^*\)-subalgebra, then we show that X must be a Stonean space. Thus Kadison’s result can never be extended beyond \(AW^*\)-algebras.

MSC:

46L05 General theory of \(C^*\)-algebras
47B38 Linear operators on function spaces (general)
54G05 Extremally disconnected spaces, \(F\)-spaces, etc.
46J10 Banach algebras of continuous functions, function algebras
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
55R25 Sphere bundles and vector bundles in algebraic topology
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References:

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