Uniformly hyperfinite algebra

Definition

A UHF C*-algebra is the direct limit of an inductive system {An, φn} where each An is a finite-dimensional full matrix algebra and each φn : An → A**n+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 rk**n = k**n + 1 for some integer r and

:\phi_n (a) = a \otimes I_r,

where Ir is the identity in the r × r matrices. The sequence ...kn|k**n + 1|k**n + 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 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 tp in δ(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 δ(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.

References

References

1. Rørdam, M.. (2000). "An Introduction to K-Theory for C*-Algebras". Cambridge University Press.
2. Glimm, James G.. (1 February 1960). "On a certain class of operator algebras". Transactions of the American Mathematical Society.
3. Davidson, Kenneth. (1997). "C*-Algebras by Example". Fields Institute.