Universal AF-algebras

Abstract

We study the approximately finite-dimensional (AF) C⁎-algebras that appear as inductive limits of sequences of finite-dimensional C⁎-algebras and left-invertible embeddings. We show that there is such a separable AF-algebra AF which is a split-extension of any finite-dimensional C⁎-algebra and has the property that any separable AF-algebra is isomorphic to a quotient of AF. Equivalently, by Elliott's classification of separable AF-algebras, there are surjectively universal countable scaled (or with order-unit) dimension groups. This universality is a consequence of our result stating that AF is the Fraïssé limit of the category of all finite-dimensional C⁎-algebras and left-invertible embeddings.With the help of Fraïssé theory we describe the Bratteli diagram of AF and provide conditions characterizing it up to isomorphisms. AF belongs to a class of separable AF-algebras which are all Fraïssé limits of suitable categories of finite-dimensional C⁎-algebras, and resemble C(2N) in many senses. For instance, they have no minimal projections, tensorially absorb C(2N) (i.e. they are C(2N)-stable) and satisfy similar homogeneity and universality properties as the Cantor set.