Noncommutative Lattices and the Algebra of their Continuous Functions

Recently a new kind of approximation to continuum topological spaces has been introduced, the approximating spaces being partially ordered sets (posets) with a finite or at most a countable number of points. The partial order endows a poset with a nontrivial non-Hausdorff topology. Their ability to reproduce important topological information of the continuum has been the main motivation for their use in quantum physics. Posets are truly noncommutative spaces, or {\it noncommutative lattices}, since they can be realized as structure spaces of noncommutative $C^*$-algebras. These noncommutative algebras play the same role of the algebra of continuous functions ${\cal C}(M)$ on a Hausdorff topological space $M$ and can be thought of as algebras of operator valued functions on posets. In this article, we will review some mathematical results that establish a duality between finite posets and a certain class of C$^*$-algebras. We will see that the algebras in question are all postliminal approximately finite dimensional (AF) algebras.