Uniformly hyperfinite algebra

In operator algebras, a uniformly hyperfinite, or UHF, algebra is one that is the closure, in the appropriate topology, of an increasing union of finite dimentional full matrix algebras.

C*-algebras

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

$A={\overline {\cup _{n}A_{n}}}.$

If

$A_{n}\simeq M_{k_{n}}(\mathbb {C} ),$

then k_n = r k_n + 1 for some integer r and

$\phi _{n}(a)=a\otimes I_{r},$

where I_r is the identity in the r × r matrices. The sequence ...k_n|k_n + 1|k_n + 2 determines a formal product

$\delta (A)=\prod _{p}p^{t_{p}}$

where each p is prime and t_p = sup {m|p_m divides k_n for some n}, possibly zero or infinite. The formal product δ(A) is said to be the super natural number corresponding to A. Glimm showed that the supernatural number is a complete invariant of UHF C*-algebras. In particular, there are uncountably many UHF C*-algebras.

One example of a UHF C*-algebra is the CAR algebra.