Involutive quantales were introduced in 7 as complete lattices equipped with a multiplication and an involution. Such structures are well known from the calculus Ž . of relations: the set R Rel X of binary relations on any set X forms an involutive quantale. However, the motivating example of an involutive quantale is the spectrum Max A of a non-commutative C*-algebra A, where Max A is the w x involutive quantale consisting of all closed linear subspaces of A. In 8 it was indeed shown that A can be reconstructed from the involutive quantale Max A, demonstrating that involutive quantales do provide a convenient algebraic invariant of non-commutative C*-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 Ž . morphisms. We also show that this class contains all quantales Q Q S of sup-preserving endomorphisms on a complete lattice S having a duality, a fortiori, all w x Ž . Hilbert quantales as defined in 7 . A special case of this is R Rel X , which can be Ž X . seen to be Q Q 2 .
Our investigation of the class of simple involutive quantales is also motivated by w x C*-algebras. It is proved in 8 that for any C*-algebra A, Max A has enough Ž Ž .. Ž . homomorphisms into the Hilbert quantales Q Q P P H , where P P H is the complete lattice of closed linear subspaces of a Hilbert space H. In this spirit, viewing Ž . involutive quantales as duals of generalized non-commutative topological spaces, * The authors gratefully acknowledge financial support from, respectively, the Natural Sciences and Engineering Research Council of Canada under Grant OGP0009134 and the Grant Agency of the Czech Republic under Grant 201r93r0950.