Some Results on Simple Quantales
by
Jan Paseka
Masaryk University

This work is intended as a step towards the development of the non-commutative topology using the approach of the theory of (involutive) quantales developed by C.J. Mulvey, J.W. Pelletier and J. Rosický. Quantales are certain partially ordered algebraic structures which generalize frames (pointless topologies) as well as various lattices of multiplicative ideals from ring theory and functional analysis (C^*-algebras, von Neumann algebras). The relations on a set, under the operation of relational composition, also form an involutive quantale.

A quantale is then a complete lattice Q with an associative binary multiplication satisfying

x · Ú i in I  xi = Ú i in I  x ·xi  and   æ è Ú i in I  xi ö ø ·x = Ú i in I  xi ·x

for all x, x_i in Q, i in I (I is a set). An element x is called right-sided (left-sided ) if x ·1 <= x (1 ·x <= x). The set of all right-sided elements is denoted by R(Q) (L(Q)). A quantale Q is called semiright (semileft) if 1 is the right (left) unit for R(Q) (L(Q)). A quantale Q is said to be semiunital if it is both semileft and semiright.

By an involutive quantale will be meant a quantale Q together with an involution ^* satisfying, for all a_i in Q, the following ( \/ a_i)^* = \/ a_i^*.

The aim of the presented lecture is the study of the notion of simplicity in semiunital (involutive) quantales. Our main result is a characterization of simple (involutive) semiunital quantales i.e. those (involutive) semiunital quantales having only trivial (involutive) quotients. Note that, 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 give a natural construction of simple (involutive) semiunital quantales arising from the category of complete semilattices (ortholattices). Some connections to the theory of C^*-algebras are established. A distinction betweeen simple semiunital quantales and simple involutive semiunital quantales is presented.

Date received: June 24, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # caai-84.