On Quantales and Spectra of C*-Algebras |
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Authors: | David Kruml Joan Wick Pelletier Pedro Resende Jiří Rosický |
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Institution: | (1) Department of Algebra and Geometry, Faculty of Sciences, Masaryk University, Janákovo nám. 2a, 66295 Brno, Czech Republic;(2) Department of Mathematics and Statistics, York University, North York, Ontario, M3J 1P3, Canada;(3) Departamento de Matemática, Instituto Superior Técnico Av. Rovisco Pais, 1049-001 Lisboa, Portugal |
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Abstract: | We study properties of the quantale spectrum MaxA of an arbitrary unital C*-algebra A. In particular we show that the spatialization of MaxA 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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Keywords: | noncommutative space C*-algebra noncommutative spectrum spatial quantale |
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