Examples
By definition, uniformly hyperfinite algebras are AF and unital. Their dimension groups are the countable subgroups of R. For example, for the 2 × 2 matrices M2, K0(M2) is Z, the rational numbers of the form a/2. The scale is Γ(M2) = Z ∩ = . For the CAR algebra A, K0(A) is the dyadic rationals with scale K0(A) ∩, with 1 = . All such groups are simple, in a sense appropriate for ordered groups. Thus UHF algebras are simple C*-algebras. In general, the groups which are not dense are the dimension groups of Mk for some k.
Commutative C*-algebras, which were characterized by Gelfand, are AF precisely when the spectrum is totally disconnected. The continuous functions C(X) on the Cantor set X is one such example.
Read more about this topic: Approximately Finite Dimensional C*-algebra
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