Uniformly Hyperfinite Algebra - Definition and Classification

Definition and Classification

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. Suppressing the connecting maps, one can write

If

then r k_n = k_{n + 1} for some integer r and

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

where each p is prime and t_p = sup {m | pm divides k_n 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_δ(A). A UHF algebra is said to be of infinite type if each t_p in δ(A) is 0 or ∞.

In the language of K-theory, each supernatural number

specifies an additive subgroup of R that is the rational numbers of the type n/m where m formally divides δ(A). This group is the K₀ group of A.