Uniformly hyperfinite algebra

Last updated April 01, 2019

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.

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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

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A={\overline {\cup _{n}A_{n}}}.

Classification

If

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.^[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.

In mathematics, the supernatural numbers, sometimes called generalized natural numbers or Steinitz numbers, are a generalization of the natural numbers. They were used by Ernst Steinitz in 1910 as a part of his work on field theory.

James Gilbert Glimm is an American mathematician, former president of the American Mathematical Society, and distinguished professor at Stony Brook University. He has made many contributions in the areas of pure and applied mathematics.

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

In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras. It can be seen as the study of certain kinds of invariants of large matrices.

\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. ^[1]

In mathematics, operator K-theory is a noncommutative analogue of topological K-theory for Banach algebras with most applications used for C*-algebras.

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.

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References

1 2 Rørdam, M.; Larsen, F.; Laustsen, N.J. (2000). An Introduction to K-Theory for C*-Algebras. Cambridge: Cambridge University Press. ISBN 0521789443.
↑ 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.
↑ Davidson, Kenneth (1997). C*-Algebras by Example. Fields Institute. pp. 166, 218–219, 234. ISBN 0-8218-0599-1.

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[Rordam00-1] 1 2 Rørdam, M.; Larsen, F.; Laustsen, N.J. (2000). An Introduction to K-Theory for C*-Algebras. Cambridge: Cambridge University Press. ISBN 0521789443.

[glimm60-2] 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.

[Davidson97-3] Davidson, Kenneth (1997). C*-Algebras by Example. Fields Institute. pp. 166, 218–219, 234. ISBN 0-8218-0599-1.