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:<math>A_n \simeq M_{k_n} (\mathbb C),</math>
then ''
:<math>\phi_n (a) = a \otimes I_r,</math>
where ''I<sub>r</sub>'' is the identity in the ''r'' × ''r'' matrices. The sequence ...''k<sub>n</sub>''|''k''<sub>''n'' + 1</sub>|''k''<sub>''n'' + 2</sub>... determines a formal product
:<math>\delta(A) = \prod_p p^{t_p}</math>
where each ''p'' is prime and ''t<sub>p</sub>'' = sup {''m'' | ''p<sup>m</sup>'' 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''.<ref name=Rordam00>{{cite book|last=Rørdam|first=M.|last2=Larsen|first2=F.|last3=Laustsen|first3=N.J.|title=An Introduction to K-Theory for C*-Algebras|year=2000|publisher=Cambridge University Press|___location=Cambridge|isbn=0521789443}}</ref> [[James Glimm|Glimm]] showed that the supernatural number is a complete invariant of UHF C*-algebras.<ref name=glimm60>{{cite journal|last=Glimm|first=James G.|title=On a certain class of operator algebras|journal=Transactions of the American Mathematical Society|date=1 February 1960|volume=95|issue=2|pages=318–340|doi=10.1090/S0002-9947-1960-0112057-5|url=http://www.ams.org/journals/tran/1960-095-02/S0002-9947-1960-0112057-5/S0002-9947-1960-0112057-5.pdf|accessdate=2 March 2013|doi-access=free}}</ref> In particular, there are uncountably many isomorphism classes of 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 '''Q''' that is the rational numbers of the type ''n''/''m'' where ''m'' formally divides ''δ''(''A''). This group is the [[Operator K-theory|''K''<sub>0</sub> group]] of ''A''. <
== CAR algebra ==
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