Geodesic
Real Af C*-Algebras With K 0 of Small Rank
Canadian journal of mathematics, Tome 41 (1989) no. 5, pp. 786-807

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A real AF C*-algebra is the norm closure of a direct limit of finite dimensional real C*-algebras (with real *-algebra maps). When we use the unadorned “AF C*-algebra”, we mean the usual complex version.Let R be a simple AF C*-algebra such that K 0(R) is free of rank 2 or 3. The problem is to find (up to Morita equivalence) all real AF C*-algebras A such that AꕕC≅R. This is closely related to the problem of finding all involutions on R [3], [10].For example, when the rank is 2, generically there are 8 such classes. The exceptional cases arise when the ratio of the two generators in K 0(R) is a quadratic (algebraic) number, and here there are 4, 5, or 8 Morita equivalence classes, the number depending largely on the behaviour of the prime 2 in the relevant algebraic number field. 
Giordano, T.; Handelman, D. E. Real Af C*-Algebras With K 0 of Small Rank. Canadian journal of mathematics, Tome 41 (1989) no. 5, pp. 786-807. doi: 10.4153/CJM-1989-036-6
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