On the quantisation of points

AbstractIn the study of quantales arising naturally in the context of C∗-algebras, Gelfand quantales have emerged as providing the basic setting. In this paper, the problem of defining the concept of point of the spectrum MaxA of a C∗-algebra A, which is one of the motivating examples of a Gelfand quantale, is considered. Intuitively, one feels that points should correspond to irreducible representations of A. Classically, the notions of topological and algebraic irreducibility of a representation are equivalent. In terms of quantales, the irreducible representations of a C∗-algebra A are shown to be captured by the notion of an algebraically irreducible representation of the Gelfand quantale MaxA on an atomic orthocomplemented sup-lattice S, defined in terms of a homomorphism of Gelfand quantales to the Hilbert quantale Q(S) of sup-preserving endomorphisms on S. This characterisation leads to a concept of point of an arbitrary Gelfand quantale Q as a map of Gelfand quantales into a Hilbert quantale Q(S), the inverse image homomorphism of which is an algebraically irreducible representation of Q on the atomic orthocomplemented sup-lattice S. The aptness of this definition of point is demonstrated by observing that in the case of locales it is exactly the classical notion of point, while the Hilbert quantale Q(S) of an atomic orthocomplemented sup-lattice S has, up to equivalence, exactly one point. In this sense, the Hilbert quantale Q(S) is considered to be a quantised version of the one-point space