On the quantisation of spaces

Abstract

In a previous paper (Pure Appl. Algebra 159 (2001) 231) a definition of point of a Gelfand quantale is given in terms of algebraically irreducible representations of the quantale on an atomic orthocomplemented sup-lattice. The definition yields the usual notion of point when applied to a locale viewed as a quantale and helps to establish that the spectrum Max A of a C ∗ -algebra A is an invariant of A. The current paper is concerned with finding a notion of spatiality in which, as intuitively expected, Max A is spatial, and for which there is an intimate connection with points in the above sense. Moreover, the approach taken relates naturally to that of Giles and Kummer in their article on non-commutative spaces (Indiana Math. J. 21 (1971) 91). Explicitly, an involutive unital quantale X is said to be spatial provided that it admits an algebraically strong right embedding into a discrete von Neumann quantale Q. It is proved that X is spatial if, and only if, it has enough points, and that, in this case, it is necessarily a Gelfand quantale. In particular, a locale is spatial as an involutive unital quantale precisely when it yields the topology of a classical topological space. A notion of quantal space is defined, and it is shown that any involutive unital quantale admits a spatialisation determined by its points, generalising that known for locales. In particular, any quantal space admits an underlying topological space. Finally, Max A is shown to be spatial, and to determine a canonical quantal space, as desired.