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 {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

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

Classification

If

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

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

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

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

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

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 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 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

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

specifies an additive subgroup of Q that is the rational numbers of the type n/m where m formally divides δ(A). This group is the K₀ group of A. ^[1]

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 f_n and L(H) the bounded operators on H, consider a linear map

${\displaystyle \alpha :H\rightarrow L(H)}$

with the property that

${\displaystyle \{\alpha (f_{n}),\alpha (f_{m})\}=0\quad {\mbox{and}}\quad \alpha (f_{n})^{*}\alpha (f_{m})+\alpha (f_{m})\alpha (f_{n})^{*}=\langle f_{m},f_{n}\rangle I.}$

The CAR algebra is the C*-algebra generated by

${\displaystyle \{\alpha (f_{n})\}\;.}$

The embedding

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

can be identified with the multiplicity 2 embedding

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

Therefore, the CAR algebra has supernatural number 2^∞.^[3] This identification also yields that its K₀ group is the dyadic rationals.