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In [[mathematics]], particularly in the theory of [[C*-algebras]], a '''uniformly hyperfinite''', or '''UHF''', algebra is a C*-algebra that can be written as the closure, in the [[Operator norm|norm topology]], of an increasing union of finite-dimensional full [[matrix ring|matrix algebras]].
== Definition
A UHF C*-algebra is the [[direct limit]] of an inductive system {''A<sub>n</sub>'', ''φ<sub>n</sub>''} where each ''A<sub>n</sub>'' is a finite-dimensional full matrix algebra and each ''φ<sub>n</sub>'' : ''A<sub>n</sub>'' → ''A''<sub>''n''+1</sub> is a unital embedding. Suppressing the connecting maps, one can write
:<math>A = \overline {\cup_n A_n}.</math>
== Classification ==
If
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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''. <cite name=Rordam00 />
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
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