Prime Elements of Non-integral Quantales and their Applications

Abstract

The quantisation of the Boolean algebra 2 is given by the semi-integral regularization of the quantale of all join preserving self-maps of the chain of three elements. On this basis prime elements of quantales are identified with strong homomorphisms taking their values in the quantisation of 2. A quantale is spatial iff strong homomorphisms with values in the quantisation of 2 separate elements. Since the adjunction between locales and topological spaces has a non-commutative extension to an adjunction between quantales and many valued topological spaces, every spatial semiunital quantale can be identified with a many-valued sober space. This result is applied to the topologization of spectra of non-commutative C∗-algebras.