Consider first the special case where Γ("G") = [0, "uG"] with maximal element "uG". Suppose  Dropping to a subsequence if necessary, let  where "φ"1("u"1) = "uG" for some element "u"1. Now consider the order ideal "G"1 generated by "u"1. Because each "H"1 has the canonical order structure, "G"1 is a direct sum of Z 's (with the number of copies possible less than that in "H"1). So this gives a finite-dimensional algebra "A"1 whose dimension group is ("G"1 "G"1+, [0, "u"1]). Next move "u"1 forward by defining "u"2 = "φ"12("u"1). Again "u"2 determines a finite-dimensional algebra "A"2. There is a corresponding homomorphism "α"12 such that "α"12* = φ12. Induction gives a directed system  whose "K"0 is  with scale  This proves the special case. A similar argument applies in general. Observe that the scale is by definition a directed set. If Γ("G") = {"vk"}, one can choose "uk" ∈ Γ("G") such that "uk" ≥ "v"1 ... "vk". The same argument as above proves the theorem. By definition, uniformly hyperfinite algebras are AF and unital. Their dimension groups are the subgroups of Q. For example, for the 2 × 2 matrices "M"2, "K"0("M"2) is the group of rational numbers of the form "a"/2 for "a" in Z. The scale is Γ("M"2) = {0, ½, 1}. For the CAR algebra "A", "K"0("A") is the group of dyadic rationals with scale "K"0("A") ∩ [0, 1], with 1 = [1"A"]. 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 in Q 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.