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On quantales and spectra of C*-algebras

David Kruml·Joan Wick Pelletier·Pedro Resende·Jiri Rosicky
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Abstract

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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David Kruml, Joan Wick Pelletier, Pedro Resende, Jiri Rosicky. 2002-11-21. On quantales and spectra of C*-algebras. https://arxiv.org/abs/math/0211345

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