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where each ''p'' is prime and ''t<sub>p</sub>'' = sup {''m''|''p<sub>m</sub>'' divides ''k<sub>n</sub> '' for some ''n''}, possibly zero or infinite. The formal product ''δ''(''A'') is said to be the '''supernatural number''' corresponding to ''A''. [[James Glimm|Glimm]] showed that the supernatural number is a complete invariant of UHF C*-algebras. In particular, there are uncountably many UHF C*-algebras.
=== An example===
One example of a UHF C*-algebra is the [[CAR algebra]]. It is defined as follows: let ''H'' be a separable complex Hilbert space ''H'' with orthonormal basis ''f<sub>n</sub>'' and ''L''(''H'') the bounded operators on ''H'', consider a linear map
:<math>\alpha : H \rightarrow L(H)</math>
with the property that
:<math>
\{ \alpha(f_n), \alpha(f_m) \} = 0 \quad \mbox{and} \quad \alpha(f_n)^*\alpha(f_m) + \alpha(f_m)\alpha(f_n)^* =
\langle f_m, f_n \rangle I.
</math>
The CAR algebra is the C*-algebra generated by
:<math>\{ \alpha(f_n) \}\;.</math>
The embedding
:<math>C^*(\alpha(f_1), \cdots, \alpha(f_n)) \hookrightarrow C^*(\alpha(f_1), \cdots, \alpha(f_{n+1}))</math>
can be identified with the multiplicity 2 embedding
:<math>M_{2^n} \hookrightarrow M_{2^{n+1}}.</math>
Therefore the CAR algebra has supernatural number 2<sup>∞</sup>. This identification also yields that its [[K-theory|''K''<sub>0</sub>]] group is the [[dyadic rational]]s.
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