Revisiting the Farey AF algebra

In a recent paper, F. Boca investigates the AF algebra A associated with the Farey-Stern-Brocot sequence. We show that A coincides with the AF algebra m1 introduced by the present author in 1988. As proved in that paper (Adv. Math., vol. 68.1), the K0-group of A is the lattice-ordered abelian group M1 of piecewise linear functions on the unit interval, each piece having integer coefficients, with the constant 1 as the distinguished order unit. Using the elementary properties of M1 we can give short proofs of several results in Boca's paper. We also prove many new results: among others, A is a *-subalgebra of Glimm universal algebra, tracial states of A are in one-one correspondence with Borel probability measures on the unit real interval, all primitive ideals of A are essential. We describe the automorphism group of A. For every primitive ideal I of A we compute K0(I) and K0(A/I).