Revisiting the Farey AF Algebra

Abstract

In a recent paper, F. Boca investigates the AF algebra \({{\mathfrak{A}}}\) associated with the Farey-Stern-Brocot sequence. We show that \({{\mathfrak{A}}}\) coincides with the AF algebra \({{\mathfrak{M_{1}}}}\) introduced by the present author in 1988. As proved in that paper (Adv. Math., vol.68.1), the K0-group of \({\mathfrak{A}}\) is the lattice-ordered abelian group \({\mathcal{M}_{1}}\) 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 \({\mathcal{M}_{1}}\) we can give short proofs of several results in Boca’s paper. We also prove many new results: among others, \({{\mathfrak{A}}}\) is a *-subalgebra of Glimm universal algebra, tracial states of \({{\mathfrak{A}}}\) are in one-one correspondence with Borel probability measures on the unit real interval, all primitive ideals of \({{\mathfrak{A}}}\) are essential. We describe the automorphism group of \({{\mathfrak{A}}}\) . For every primitive ideal I of \({{{\mathfrak{A}}}}\) we compute K0(I) and \({{K_{0}(\mathfrak{A}/I)}}\).