Simple Involutive Quantales

Abstract

Involutive quantales were introduced in [7] as complete lattices equipped with a multiplication and an involution. Such structures are well known from the calculus of relations: the set Rel(X) of binary relations on any setXforms an involutive quantale. However, the motivating example of an involutive quantale is the spectrum MaxAof a non-commutativeC*-algebraA, where MaxAis the involutive quantale consisting of all closed linear subspaces ofA. In [8] it was indeed shown thatAcan be reconstructed from the involutive quantale MaxA, demonstrating that involutive quantales do provide a convenient algebraic invariant of non-commutativeC*-algebras.The aim of this paper is to study involutive quantales from the algebraic point of view. Our main result is the characterization of simple involutive quantales, that is, involutive quantales on which the only surjections are isomorphisms or constant morphisms. We also show that this class contains all quantales Q(S) of sup-preserving endomorphisms on a complete latticeShaving a duality, a fortiori, all Hilbert quantales as defined in [7]. A special case of this is Rel(X), which can be seen to be Q(2X).Our investigation of the class of simple involutive quantales is also motivated byC*-algebras. It is proved in [8] that for anyC*-algebraA, MaxAhas enough homomorphisms into the Hilbert quantales Q(P(H)), where P(H) is the complete lattice of closed linear subspaces of a Hilbert spaceH. In this spirit, viewing involutive quantales as duals of generalized (non-commutative) topological spaces, simple involutive quantales correspond to spaces having only trivial subspaces, that is, to points. We begin a study of those involutive quantales which have enough homomorphisms into simple involutive quantales, noting their “spatial” character. Such a study has algebraic antecedents in the early work [4] and the more recent work [1] which studied certain lattices (noncomplete involutive quantales) having enough homomorphisms into simple involutive quantales of the form Rel(X).