# Recognizing the Farey–Stern–Brocot AF algebra

### Daniele Mundici

Università degli Studi di Firenze, Italy

## Abstract

In his 2008 paper published in the Canadian Journal of Mathematics, F. Boca investigates an AF algebra $A$, whose Bratteli diagram arises from the Farey–Stern–Brocot sequence. It turns out that $A$ coincides with the AF algebra $M_{1}$ introduced in 1988 by the present author in a paper published in Advances in Mathematics. We give a procedure to recognize $A$ among all ﬁnitely presented AF algebras whose Murray–von Neumann order of projections is a lattice. Further: (i) $A$ is a *-subalgebra of Glimm universal algebra; (ii) tracial states of $A$ correspond to Borel probability measures on the unit real interval; (iii) all primitive ideals of $A$ are essential; (iv) the automorphism group of $A$ has exactly two connected components.

## Cite this article

Daniele Mundici, Recognizing the Farey–Stern–Brocot AF algebra. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 20 (2009), no. 4, pp. 327–338

DOI 10.4171/RLM/549