One of the principal reasons for the introduction of the concept of quantale was to find a categorical context, generalising to the non-commutative case that of locales, within which the insights of Giles and Kummer into the spectral representation of C*-algebras might be placed. Consideration of their ideas led to the concept of the spectrum Max A of any C*-algebra A , functorial on the category of C*-algebras and admitting a Gelfand-Naimark representation exactly generalising that in the commutative case. The question remained of whether this spectrum Max A could be considered in some sense to be a quantal space. Approaching from the classical belief that its points should be the equivalence classes of irreducible representations, one may obtain a concept of a point of an involutive quantale which satisfies this criterion, yet extends that of a point of a locale. Generalising again the situation for locales, one may characterise those involutive quantales that are spatial as those having enough points, arriving finally at the concept of a quantal space.
Christopher Mulvey (University of Sussex, UK)
February 09, 2001