Universal AF-algebras

We study the approximately finite-dimensional (AF) $C^{\ast }$-algebras that appear as inductive limits of sequences of finite-dimensional $C^{\ast }$-algebras and left-invertible embeddings. We show that there is such a separable AF-algebra ${\mathcal{A}}_{\mathfrak{F}}$ with the property that any separable AF-algebra is isomorphic to a quotient of ${\mathcal{A}}_{\mathfrak{F}}$. 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 $\mathcal A_\mathfrak{F}$ is the Fra"\i ss'e limit of the category of all finite-dimensional $C^{\ast }$-algebras and left-invertible embeddings. With the help of Fra"\i ss'e theory we describe the Bratteli diagram of ${\mathcal{A}}_{\mathfrak{F}}$ and provide conditions characterizing it up to isomorphisms. ${\mathcal{A}}_{\mathfrak{F}}$ belongs to a class of separable AF-algebras which are all Fra"\i ss'e limits of suitable categories of finite-dimensional $C^{\ast }$-algebras, and resemble $C(2^{\mathbb{N}})$ in many senses. For instance, they have no minimal projections, tensorially absorb $C(2^{\mathbb{N}})$ (i.e. they are $C(2^{\mathbb{N}})$-stable) and satisfy similar homogeneity and universality properties as the Cantor set.