On quantales and spectra of C*-algebras

Details On quantales and spectra of C*-algebras On quantales and spectra of C*-algebras

2002-11-21

arxiv.org/abs/math/0211345v1

Applied Categorical Structures 11 (2003) 543-560

Mathematics

Operator Algebras

22 pages. Will appear in Applied Categorical Structures

Scientific paper

We study properties of the quantale spectrum Max A of an arbitrary unital C*-algebra A. In particular we show that the spatialization of Max A with respect to one of the notions of spatiality in the literature yields the locale of closed ideals of A when A is commutative. We study under general conditions functors with this property, in addition requiring that colimits be preserved, and we conclude in this case that the spectrum of A necessarily coincides with the locale of closed ideals of the commutative reflection of A. Finally, we address functorial properties of Max, namely studying (non-)preservation of limits and colimits. Although Max is not an equivalence of categories, therefore not providing a direct generalization of Gelfand duality to the noncommutative case, it is a faithful complete invariant of unital C*-algebras.

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