Uniformly hyperfinite algebra


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.

Definition

A UHF C*-algebra is the direct limit of an inductive system where each An is a finite-dimensional full matrix algebra and each φn : An → An+1 is a unital embedding. Suppressing the connecting maps, one can write

Classification

If
then rkn = kn + 1 for some integer r and
where Ir is the identity in the r × r matrices. The sequence...kn|kn + 1|kn + 2... determines a formal product
where each p is prime and tp = sup, possibly zero or infinite. The formal product δ 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 is finite, then A is the full matrix algebra Mδ. A UHF algebra is said to be of infinite type if each tp in δ is 0 or ∞.
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. This group is the K0 group of A.

CAR algebra

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 fn and L 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∞. This identification also yields that its K0 group is the dyadic rationals.