On quantales that classify C*-algebras

TL;DR

This paper proves that the functor Max classifies unital C*-algebras up to *-isomorphism via involutive quantales, but also shows limitations where quantale isomorphisms do not correspond to algebra isomorphisms.

Contribution

It provides a proof that Max classifies unital C*-algebras and reveals cases where quantale isomorphisms do not correspond to algebra isomorphisms.

Findings

Max classifies unital C*-algebras up to *-isomorphism.

There exist quantale isomorphisms not induced by algebra isomorphisms.

The paper establishes a stronger correspondence between algebra and quantale isomorphisms.

Abstract

The functor Max of Mulvey assigns to each unital C*-algebra A the unital involutive quantale Max A of closed linear subspaces of A, and it has been remarked that it classifies unital C*-algebras up to *-isomorphism. In this paper we provide a proof of this and of the stronger fact that for every isomorphism u : Max A -> Max B of unital involutive quantales there is a *-isomorphism u' : A -> B such that Max u' coincides with u when restricted to the left-sided elements of Max A. But we also show that isomorphisms u : Max A -> Max B may exist for which no isomorphism v : A -> B is such that Max v = u.