Rendiconti Lincei - Matematica e Applicazioni
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Published online: 2009-12-23
Recognizing the Farey–Stern–Brocot AF algebraDaniele Mundici (1) Università degli Studi di Firenze, Italy
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.
Keywords: Approximately ﬁnite dimensional algebra, Elliott classiﬁcation, Grothendieck group, lattice-ordered group, Farey sequence
Mundici Daniele: Recognizing the Farey–Stern–Brocot AF algebra. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 20 (2009), 327-338. doi: 10.4171/RLM/549