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On the quantisation of points. (English) Zbl 0983.18007
Quantales arose as a non-commutative generalization of the notion of locales (frames) and, in this article, the authors continue their work exploring that aspect of quantale theory concerned with the key example of closed ideals of a \(C^*\)-algebra. The article under review had its genesis in the article “A quantisation of the calculus of relations,” [C. J. Mulvey and J. W. Pelletier, in: Category Theory 1991, Am. Math. Soc., CMS Conf. Proc. 13, 345-360 (1992; Zbl 0793.06008)], where the central notion of Gelfand quantale was first introduced. Using this as a starting point, the authors attempt to define the notion of a point of a Gelfand quantale, generalizing the corresponding notion of point of a locale. The authors are motivated by the idea that the points of Max \(A\), the spectrum of a \(C^*\)-algebra \(A\), should somehow correspond to the irreducible representations of \(A\). Representations of \(C^*\)-algebras are studied via homomorphisms of Gelfand quantales and central to the notion of point for a Gelfand quantale is the study of the Hilbert quantale \({\mathcal Q}(S)\) of sup-preserving mappings from an ortho-complemented sup-lattice \(S\) to itself. Utilizing the analogy of a representation of a \(C^*\)-algebra \(A\) on a Hilbert space \(H\) with a representation of a Gelfand quantale \(Q\) on an ortho-complemented sup-lattice \(S\) (a Gelfand quantale homomorphism \(Q\to {\mathcal Q}(S))\), the authors proceed to investigate these ideas and then apply them to the theory of \(C^*\)-algebras.

MSC:
18F99 Categories in geometry and topology
06F07 Quantales
49L99 Hamilton-Jacobi theories
46L30 States of selfadjoint operator algebras
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