Universal AF-algebras

We study the approximately finite-dimensional (AF) C*-algebras that appear as inductive limits of sequences of finitedimensional C*-algebras and left-invertible embeddings. We show that there is such a separable AF-algebra A(F) 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 A(F). 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 A(F) is the Fraisse limit of the category of all finite-dimensional C*-algebras and left-invertible embeddings. With the help of Fraisse theory we describe the Bratteli diagram A(F) of and provide conditions characterizing it up to isomorphisms. A(F) belongs to a class of separable AF-algebras which are all Fraisse limits of suitable categories of finite-dimensional C*-algebras, and resemble C(2(N)) in many senses. For instance, they have no minimal projections, tensorially absorb C(2(N)) (i.e. they are C(2(N))-stable) and satisfy similar homogeneity and universality properties as the Cantor set. (C) 2020 Elsevier Inc. All rights reserved.