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:<math>\delta(A) = \prod_p p^{t_p}</math>
[[Category:C*-algebras]]▼
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.
If ''δ''(''A'') is finite, then ''A'' is the full matrix algebra ''M''<sub>''δ''(''A'')</sub>. A UHF algebra is said to be of ''infinite type'' if each ''t<sub>p</sub>'' in ''δ''(''A'') is 0 or ∞.
In the language of [[K-theory]], each supernatural number
:<math>\delta(A) = \prod_p p^{t_p}</math>
specifies an additive subgroup of '''R''' that is the rational numbers of the type ''n''/''m'' where ''m'' formally divides ''δ''(''A''). This group is called the ''K''<sub>0</sub> group of ''A''.
=== An example===
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:<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
▲[[Category:C*-algebras]]
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