Revisiting the Farey AF Algebra |
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Authors: | Daniele Mundici |
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Institution: | 1.Department of Mathematics “Ulisse Dini”,University of Florence,Florence,Italy |
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Abstract: | In a recent paper, F. Boca investigates the AF algebra
\mathfrakA{{\mathfrak{A}}} associated with the Farey-Stern-Brocot sequence. We show that
\mathfrakA{{\mathfrak{A}}} coincides with the AF algebra
\mathfrakM1{{\mathfrak{M_{1}}}} introduced by the present author in 1988. As proved in that paper (Adv. Math., vol.68.1), the K
0-group of
\mathfrakA{\mathfrak{A}} is the lattice-ordered abelian group M1{\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 M1{\mathcal{M}_{1}} we can give short proofs of several results in Boca’s paper. We also prove many new results: among others,
\mathfrakA{{\mathfrak{A}}} is a *-subalgebra of Glimm universal algebra, tracial states of
\mathfrakA{{\mathfrak{A}}} are in one-one correspondence with Borel probability measures on the unit real interval, all primitive ideals of
\mathfrakA{{\mathfrak{A}}} are essential. We describe the automorphism group of
\mathfrakA{{\mathfrak{A}}} . For every primitive ideal I of
\mathfrakA{{{\mathfrak{A}}}} we compute K
0(I) and
K0(\mathfrakA/I){{K_{0}(\mathfrak{A}/I)}}. |
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Keywords: | |
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