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 norm topology, of an increasing union of finite-dimensional full matrix algebras.
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
If
then rk<sub>n</sub> = k<sub>n + 1</sub> for some integer r and
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
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. Glimm showed that the supernatural number is a complete invariant of UHF C*-algebras. 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 âÂÂ.
In the language of K-theory, each supernatural number
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 K<sub>0</sub> group of A.
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
with the property that
The CAR algebra is the C*-algebra generated by
The embedding
can be identified with the multiplicity 2 embedding
Therefore, the CAR algebra has supernatural number 2<sup>âÂÂ</sup>. This identification also yields that its K<sub>0</sub> group is the dyadic rationals.