Recognizing the Farey–Stern–Brocot AF algebra

Abstract

In his 2008 paper published in the Canadian Journal of Mathematics, F. Boca investigates an AF algebra $\mathfrak{A}$, whose Bratteli diagram arises from the Farey–Stern–Brocot sequence. It turns out that $\mathfrak{A}$ coincides with the AF algebra ${\mathfrak{M}}_1$ introduced in 1988 by the present author in a paper published in Advances in Mathematics. We give a procedure to recognize $\mathfrak{A}$ among all finitely presented AF algebras whose Murray–von Neumann order of projections is a lattice. Further: (i) $\mathfrak{A}$ is a *-subalgebra of Glimm universal algebra; (ii) tracial states of $\mathfrak{A}$ correspond to Borel probability measures on the unit real interval; (iii) all primitive ideals of $\mathfrak{A}$ are essential; (iv) the automorphism group of $\mathfrak{A}$ has exactly two connected components.