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 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
A=\overline{\cupnAn}.
If
An\simeq
| M | |
| kn |
(C),
then rkn = kn + 1 for some integer r and
\phin(a)=a ⊗ Ir,
where Ir is the identity in the r × r matrices. The sequence ...kn|kn + 1|kn + 2... determines a formal product
\delta(A)=\prodp
| tp | |
| p |
where each p is prime and tp = sup, possibly zero or infinite. The formal product δ(A) is said to be the supernatural number corresponding to A.[1] Glimm showed that the supernatural number is a complete invariant of UHF C*-algebras.[2] In particular, there are uncountably many isomorphism classes of UHF C*-algebras.
If δ(A) is finite, then A is the full matrix algebra Mδ(A). A UHF algebra is said to be of infinite type if each tp in δ(A) is 0 or ∞.
In the language of K-theory, each supernatural number
\delta(A)=\prodp
| tp | |
| p |
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 K0 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 fn and L(H) the bounded operators on H, consider a linear map
\alpha:H → L(H)
with the property that
\{\alpha(fn),\alpha(fm)\}=0 and
| *\alpha(f | |
| \alpha(f | |
| m) |
+\alpha(fm)\alpha(f
| * | |
| n) |
=\langlefm,fn\rangleI.
The CAR algebra is the C*-algebra generated by
\{\alpha(fn)\} .
The embedding
| *(\alpha(f | |
| C | |
| 1), |
… ,\alpha(fn))\hookrightarrow
| *(\alpha(f | |
| C | |
| 1), |
… ,\alpha(fn+1))
can be identified with the multiplicity 2 embedding
| M | |
| 2n |
\hookrightarrow
| M | |
| 2n+1 |
.
Therefore, the CAR algebra has supernatural number 2∞.[3] This identification also yields that its K0 group is the dyadic rationals.
. Kenneth Davidson (mathematician). C*-Algebras by Example. 1997. Fields Institute. 0-8218-0599-1. 166, 218–219, 234.