Recognizing the Farey-Stern-Brocot AF algebra,

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 M1 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.