On the quantisation of spaces
Central to the theory of locales has been the connection between the notions of a point of a locale and that of a spatial locale. As the theory of quantales is on one level an attempt to generalize the theory of locales [see the reviewer's book, Quantales and their applications. Pitman Research Notes, Vol. 234. Harlow: Longman Scientific and Technical (1990; Zbl 0703.06007)], it is natural to try to develop a theory of points and spatiality in the context of quantales. This article is an extension of the fundamental work of the authors [``On the quantisation of points'', J. Pure Appl. Algebra 159, 231-295 (2001; Zbl 0983.18007)]. The setting is that of involutive unital quantales and one of the main goals is to indicate how the concepts of point and space for quantales provide the appropriate framework for looking at the spectral theory of \(C^*\)-algebras, one of the original motivations of quantale theory. After a useful review of concepts including those of involutive, Gelfand, Hilbert and von Neumann quantales from their earlier papers, they develop the theory of discrete von Neumann quantales as a prelude to defining an involutive unital quantale to be spatial provided that it admits an algebraically strong right embedding into a discrete von Neumann quantale. They proceed to prove many results including a characterization of such spatial quantales as well as the desired result that an involutive unital quantale is spatial iff it has enough points. They also show that the quantalic notion of spatiality agrees with that of spatiality for locales, when the latter are viewed as involutive, unital quantales. And finally it is shown that things work as expected for the case of the spectrum, \(\text{Max }A\), of a \(C^*\)-algebra \(A\).
scientific article; zbMATH DE number 1839072
scientific article