Noncommutative Lattices and the Algebras 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 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 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.