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On quantales and spectra of C*-algebras
by
Pedro Resende
Instituto Superior Técnico
Coauthors: D. Kruml (Masaryk Univ., Brno, Czech Republic), J. Pelletier (York Univ., Toronto, Canada), J. Rosicky (Masaryk Univ., Brno, Czech Republic)
In this talk, which is based on joint work with D. Kruml, J. Pelletier, and J. Rosický, I present three intertwined observations about the extent to which quantales may provide a notion of spectrum for C*-algebras that yields a generalization of Gelfand duality to the case of not necessarily commutative C*-algebras, where duality is meant in the categorical sense. Given a C*-algebra A, the spectrum of A is, following Mulvey, the quantale Max A of closed subspaces of A. The first observation addresses the issue of whether Max A is a spatial quantale according to the notion of spatiality proposed by Pelletier and Rosicky and further developed by Paseka and Kruml. The answer is negative but suggests that the spatialization of Max A should, in the commutative case, coincide with the locale of closed ideals of A. However, the functorial properties of such a notion of spatialization are unsatisfactory, and the second observation is that if we impose on a functor from C*-algebras to quantales that it should coincide with the locale of closed ideals in the commutative case and also that it should preserve colimits leads, under general conditions, to a functor which is very different from Max. However, Max A is spatial according to another notion of spatiality recently put forward by Mulvey and Pelletier, and thus it is worth studying the functor Max itself. The third observation is that Max preserves neither limits nor colimits, and thus it is not part of an equivalence of categories, but in part this is inessential because it is both faithful and a complete invariant of C*-algebras.
Date received: March 23, 2001
Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafw-68.