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Revisiting the Farey AF algebra. (English) Zbl 1269.46041

Summary: In a recent paper, F. P. Boca [Can. J. Math. 60, No. 5, 975–1000 (2008; Zbl 1158.46039)] investigated 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 in [the author, Adv. Math. 68, No. 1, 23–39 (1988; Zbl 0678.06008)]. As proved in that paper, the \(K _{0}\)-group of \(\mathfrak A\) is the lattice-ordered abelian group \(M_{1}\) 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 \(M_{1}\) 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 \(K _{0}(I)\) and \(K_{0}(\mathfrak A/I)\).

MSC:

46L35 Classifications of \(C^*\)-algebras
46L05 General theory of \(C^*\)-algebras
46L80 \(K\)-theory and operator algebras (including cyclic theory)
46M40 Inductive and projective limits in functional analysis
37E05 Dynamical systems involving maps of the interval
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