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

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

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

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

where 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.{{cite book|last=Rørdam|first=M.|last2=Larsen|first2=F.|last3=Laustsen|first3=N.J.|title=An Introduction to K-Theory for C*-Algebras|year=2000|publisher=Cambridge University Press|location=Cambridge|isbn=0521789443}} Glimm showed that the supernatural number is a complete invariant of UHF C*-algebras.{{cite journal|last=Glimm|first=James G.|title=On a certain class of operator algebras|journal=Transactions of the American Mathematical Society|date=1 February 1960|volume=95|issue=2|pages=318–340|doi=10.1090/S0002-9947-1960-0112057-5|url=http://www.ams.org/journals/tran/1960-095-02/S0002-9947-1960-0112057-5/S0002-9947-1960-0112057-5.pdf|accessdate=2 March 2013|doi-access=free}} 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^∞.{{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 K₀ group is the dyadic rationals.

References

Category:C*-algebras