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

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

Classification

If

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

then r k_n = 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 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

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

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 f_n 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^∞.^[3] This identification also yields that its K₀ group is the dyadic rationals.

References

1. Rørdam, M.; Larsen, F.; Laustsen, N.J. (2000). An Introduction to K-Theory for C*-Algebras. Cambridge: Cambridge University Press. ISBN 0521789443.
2. Glimm, James G. (1 February 1960). "On a certain class of operator algebras" (PDF). Transactions of the American Mathematical Society 95 (2): 318–340. doi:10.1090/S0002-9947-1960-0112057-5. Retrieved 2 March 2013.
3. Davidson, Kenneth (1997). C*-Algebras by Example. Fields Institute. pp. 166,218–219,234. ISBN 0-8218-0599-1.