Joan Wick Pelletier - Points and simplicity in quantales

The formulation of the notion of point of a quantale has been at the heart of several papers by the author and others. One method of approaching the question in the context of involutive Gelfand quantales is to extrapolate from the situation of the spectrum of a -algebra A. Here the intuitive feeling that points should be irreducible representations leads to the definition of point as an irreducible representation , where is the Hilbert quantale on the complete atomic orthomodular lattice S. This definition is validated by the fact that admits no points in this sense save the identity. Alternatively, the notion of point of a quantale can derive from that of a simple quantale, that is, a quantale having only trivial quotients. Recent characterizations of simple quantales show that the class of simple involutive quantales properly contains the above quantales . The different notions of points are investigated and compared.