Involutive quantales and quantale-enriched involutive topological spaces

Abstract

In this paper, we provide a comprehensive analysis of involutive quantales, with a particular focus on quantic frames. We extend the axiomatic foundations of quantale-enriched topological spaces to include closure under the anti-homomorphic involution, facilitating a balanced topologization of the spectrum of unital $C^*$-algebras that encompasses both closed right and left ideals through the concept of quantic frames. Specifically, certain subspaces of pure states are identified as strongly Hausdorff separated quantale-enriched involutive topological spaces.