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 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

:$A = \overline .$

Classification

If

:$A_n \simeq M_ (\mathbb C),$

then ''rk''_''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

:$\delta(A) = \prod_p p^$

where each ''p'' is prime and ''t_p'' = sup , 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

:$\delta(A) = \prod_p p^$

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 ''K''₀ 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

:$\ = 0 \quad \mbox \quad \alpha(f_n)^*\alpha(f_m) + \alpha(f_m)\alpha(f_n)^* = \langle f_m, f_n \rangle I.$

The CAR algebra is the C*-algebra generated by

:$\\;.$

The embedding

:$C^*(\alpha(f_1), \cdots, \alpha(f_n)) \hookrightarrow C^*(\alpha(f_1), \cdots, \alpha(f_))$

can be identified with the multiplicity 2 embedding

:$M_ \hookrightarrow M_.$

Therefore, the CAR algebra has supernatural number 2^∞.{{cite book, last=Davidson, first=Kenneth, authorlink=Kenneth Davidson (mathematician), title=C*-Algebras by Example, year=1997, publisher=Fields Institute, isbn=0-8218-0599-1, pages=166, 218–219, 234 This identification also yields that its ''K''₀ group is the dyadic rationals.

References

C*-algebras