On quantales and spectra of C*-algebras

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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.
22 pages. Will appear in Applied Categorical Structures

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Operator Algebras, Category Theory, 46L85 (Primary) 06F07, 46M15 (Secondary)

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http://salesiana.dossiersoluciones.com/handle/123456789/65746

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/math/0211345
 http://arxiv.org/abs/math/0211345
 Applied Categorical Structures 11 (2003) 543-560

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arXiv