NONCOMMUTATIVE LATTICES AND THE ALGEBRAS OF THEIR CONTINUOUS-FUNCTIONS

Recently a new kind of approximation to continuum topological spaces h as been introduced, the approximating spaces being partially ordered s ets (posets) with a finite or at most a countable number of points. Th e partial order endows a poset with a nontrivial non-Hausdorff topolog y. Their ability to reproduce important topological information of the continuum has been the main motivation for their use in quantum physi cs. Posets are truly noncommutative spaces, or noncommutative lattices , since they can be realized as structure spaces of noncommutative C- algebras. These noncommutative algebras play the same role as the alge bra of continuous functions C(M) on a Hausdorff topological space M an d can be thought of as algebras of operator valued functions on posets . In this article, we will review some mathematical results that estab lish a duality between finite posets and a certain class of C-algebra s. We will see that the algebras in question are all postliminal appro ximately finite dimensional (AF) algebras.