Uniformly hyperfinite algebra

In operator algebras, a uniformly hyperfinite, or UHF, algebra is one that is the closure, in the appropriate topology, of an increasing union of finite dimensional full matrix algebras.

C*-algebras

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

:$A\; =\; overline\; \{cup\_n\; A\_n\}.$

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". Glimm showed that the supernatural number is a complete invariant of UHF C*-algebras. In particular, there are uncountably many 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 R that is the rational numbers of the type "n"/"m" where "m" formally divides "δ"("A"). This group is called the "K"₀ 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 "f_n" and "L"("H") the bounded operators on "H", consider a linear map

:$alpha\; :\; H\; ightarrow\; L(H)$

with the property that

:$\{\; alpha(f\_n),\; alpha(f\_m)\; \}\; =\; 0\; quad\; mbox\{and\}\; quad\; alpha(f\_n)^*alpha(f\_m)\; +\; alpha(f\_m)alpha(f\_n)^*\; =\; langle\; f\_m,\; f\_n\; angle\; I.$

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 "K"₀ group is the dyadic rationals.