# Universal AF-algebras

## TL;DR

This paper constructs a universal separable AF-algebra as a Fraïssé limit, showing it can generate all separable AF-algebras via quotients and describing its structure through Bratteli diagrams.

## Contribution

It introduces a universal AF-algebra as a Fraïssé limit, providing a new perspective on the structure and universality of AF-algebras and their classification.

## Key findings

- Existence of a separable AF-algebra universal for all AF-algebras
- Characterization of its Bratteli diagram and isomorphism conditions
- Demonstration of its properties akin to the Cantor set, including $C(2^ ext{N})$-stability

## 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 $\mathcal A_\mathfrak{F}$ 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 $C^*$-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 $C^*$-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.