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 {An, &phi;n} where each An is a finite-dimensional full matrix algebra and each &phi;n : An &rarr; An+1 is a unital embedding. Suppressing the connecting maps, one can write


 * $$A = \overline {\cup_n A_n}.$$

Classification
If


 * $$A_n \simeq M_{k_n} (\mathbb C),$$

then rkn = kn + 1 for some integer r and


 * $$\phi_n (a) = a \otimes I_r,$$

where Ir is the identity in the r &times; r matrices. The sequence ...kn|kn + 1|kn + 2... determines a formal product


 * $$\delta(A) = \prod_p p^{t_p}$$

where each p is prime and tp = sup {m  |   pm divides kn  for some n}, possibly zero or infinite. The formal product &delta;(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 &delta;(A) is finite, then A is the full matrix algebra M&delta;(A). A UHF algebra is said to be of infinite type if each tp in &delta;(A) is 0 or ∞.

In the language of K-theory, each supernatural number


 * $$\delta(A) = \prod_p p^{t_p}$$

specifies an additive subgroup of Q that is the rational numbers of the type n/m where m formally divides &delta;(A). 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(H) the bounded operators on H, consider a linear map


 * $$\alpha : H \rightarrow L(H)$$

with the property that



\{ \alpha(f_n), \alpha(f_m) \} = 0 \quad \mbox{and} \quad \alpha(f_n)^*\alpha(f_m) + \alpha(f_m)\alpha(f_n)^* = \langle f_m, f_n \rangle I. $$

The CAR algebra is the C*-algebra generated by


 * $$\{ \alpha(f_n) \}\;.$$

The embedding


 * $$C^*(\alpha(f_1), \cdots, \alpha(f_n)) \hookrightarrow C^*(\alpha(f_1), \cdots, \alpha(f_{n+1}))$$

can be identified with the multiplicity 2 embedding


 * $$M_{2^n} \hookrightarrow M_{2^{n+1}}.$$

Therefore, the CAR algebra has supernatural number 2∞. This identification also yields that its K0 group is the dyadic rationals.