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 and classification

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 → An+1 is a unital embedding. Suppressing the connecting maps, one can write

A = \overline {\cup_n A_n}.

If

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

then r kn = kn + 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|kn + 1|kn + 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.[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 Mδ(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 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.

An example

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∞.[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". Transactions of the American Mathematical Society 95 (2): 318–318. 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.