David Kruml

Algebraic and Categorical Aspects of Quantales

Publisher Summary This chapter introduces the basic notions in theory of quantales. It describes the aspects of algebraic and categorical properties of quantales and quantale modules. This chapter also summarizes the recent developments involving representations and projectivity in quantales. Because of infinitesimal joins, quantales do not form a variety but many of the methods of universal algebra can be still used. Quantales are also applied in linear and other substructural logics and automaton theory. An important moment in the development of the theory of quantales was the realization that quantales give the semantics for propositional linear logic in the same way as Boolean algebras give semantics for classical propositional logic.

On quantales and spectra of C*-algebras

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.

Embeddings of quantales into simple quantales

Every quantale can be embedded into a simple quantale and every involutive quantale can be embedded into a somple involutive quantale. The image satisfies an analog of the von Neumann bicommutant theorem.

Girard Couples of Quantales

We introduce the concept of a Girard couple, which consists of two (not necessarily unital) quantales linked by a strong form of duality. The two basic examples of Girard couples arise in the study of endomorphism quantales and of the spectra of operator algebras. We construct, for an arbitrary sup-lattice S, a Girard quantale whose right-sided part is isomorphic to S.