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In a recent paper F Boca investigates the AF algebra documentclass 12pt minimal usepackage amsmath usepackage wasysym usepackage amsfonts usepackage amssymb usepackage amsbsy usepackage mathrsfs usepackage upgreek setlength oddsidemargin -69pt begin document mathfrak A end document associated with the Farey-Stern-Brocot sequence We show that documentclass 12pt minimal usepackage amsmath usepackage wasysym usepackage amsfonts usepackage amssymb usepackage amsbsy usepackage mathrsfs usepackage upgreek setlength oddsidemargin -69pt begin document mathfrak A end document coincides with the AF algebra documentclass 12pt minimal usepackage amsmath usepackage wasysym usepackage amsfonts usepackage amssymb usepackage amsbsy usepackage mathrsfs usepackage upgreek setlength oddsidemargin -69pt begin document mathfrak M 1 end document introduced by the present author in 1988 As proved in that paper Adv Math vol 68 1 the K0-group of documentclass 12pt minimal usepackage amsmath usepackage wasysym usepackage amsfonts usepackage amssymb usepackage amsbsy usepackage mathrsfs usepackage upgreek setlength oddsidemargin -69pt begin document mathfrak A end document is the lattice-ordered abelian group documentclass 12pt minimal usepackage amsmath usepackage wasysym usepackage amsfonts usepackage amssymb usepackage amsbsy usepackage mathrsfs usepackage upgreek setlength oddsidemargin -69pt begin document mathcal M 1 end document 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 documentclass 12pt minimal usepackage amsmath usepackage wasysym usepackage amsfonts usepackage amssymb usepackage amsbsy usepackage mathrsfs usepackage upgreek setlength oddsidemargin -69pt begin document mathcal M 1 end document we can give short proofs of several results in Boca s paper We also prove many new results among others documentclass 12pt minimal usepackage amsmath usepackage wasysym usepackage amsfonts usepackage amssymb usepackage amsbsy usepackage mathrsfs usepackage upgreek setlength oddsidemargin -69pt begin document mathfrak A end document is a -subalgebra of Glimm universal algebra tracial states of documentclass 12pt minimal usepackage amsmath usepackage wasysym usepackage amsfonts usepackage amssymb usepackage amsbsy usepackage mathrsfs usepackage upgreek setlength oddsidemargin -69pt begin document mathfrak A end document are in one-one correspondence with Borel probability measures on the unit real interval all primitive ideals of documentclass 12pt minimal usepackage amsmath usepackage wasysym usepackage amsfonts usepackage amssymb usepackage amsbsy usepackage mathrsfs usepackage upgreek setlength oddsidemargin -69pt begin document mathfrak A end document are essential We describe the automorphism group of documentclass 12pt minimal usepackage amsmath usepackage wasysym usepackage amsfonts usepackage amssymb usepackage amsbsy usepackage mathrsfs usepackage upgreek setlength oddsidemargin -69pt begin document mathfrak A end document For every primitive ideal I of documentclass 12pt minimal usepackage amsmath usepackage wasysym usepackage amsfonts usepackage amssymb usepackage amsbsy usepackage mathrsfs usepackage upgreek setlength oddsidemargin -69pt begin document mathfrak A end document we compute K0 I and documentclass 12pt minimal usepackage amsmath usepackage wasysym usepackage amsfonts usepackage amssymb usepackage amsbsy usepackage mathrsfs usepackage upgreek setlength oddsidemargin -69pt begin document K 0 mathfrak A I end document

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Revisiting the Farey AF Algebra

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with Approximately finite dimensional algebra as keyword