AF-algebra

Approximately Finite-dimensional algebra.

AF-algebras form a class of $C^*$-algebras that, on the one hand, admits an elementary construction, yet, on the other hand, exhibits a rich structure and provide examples of exotic phenomena. A (separable) $C^*$-algebra $A$ is said to be an AF-algebra if one of the following two (not obviously) equivalent conditions is satisfied (see [a1], [a2] or [a6]):

1. for every finite subset $\{a_1,\dots,a_n\}$ of $A$ and for every $\epsilon>0$ there exists a finite-dimensional sub-$C^*$-algebra $B$ of $A$ and a subset $\{b_1,\dots,b_n\}$ of $B$ with $\|a_j-b_j\|<\epsilon$ for all $j=1,\dots,n$;
2. there exists an increasing sequence $A_1\subseteq A_2\subseteq\dots$ of finite-dimensional sub-$C^*$-algebras of $A$ such that the union $\bigcup_{j=1}^\infty A_j$ is norm-dense in $A$.

Bratteli diagrams.

It follows from (an analogue of) Wedderburn's theorem (cf. Wedderburn–Artin theorem), that every finite-dimensional$C^*$-algebra is isomorphic to the direct sum of full matrix algebras over the field of complex numbers. Property 2 says that each AF-algebra is the inductive limit of a sequence $A_1\rightarrow A_2\rightarrow\dots$ of finite-dimensional $C^*$-algebras, where the connecting mappings $A_j\rightarrow A_{j+1}$ are ${}^*$-preserving homomorphisms. If two such sequences $A_1\rightarrow A_2\rightarrow\dots$ and $B_1\rightarrow B_2\rightarrow\dots$ define isomorphic AF-algebras, then already the algebraic inductive limits of the two sequences are isomorphic (as algebras over $\mathbb C$).

All essential information of a sequence $A_1\rightarrow A_2\rightarrow\dots$ of finite-dimensional $C^*$-algebras with connecting mappings can be expressed in a so-called Bratteli diagram. The Bratteli diagram is a graph, divided into rows, whose vertices in the $j$th row correspond to the direct summands of $A_j$ isomorphic to a full matrix algebra, and where the edges between the $j$th and the $(j+1)$st row describe the connecting mapping $A_j\rightarrow A_{j+1}$. By the facts mentioned above, the construction and also the classification of AF-algebras can be reduced to a purely combinatorial problem phrased in terms of Bratteli diagrams. (See [a2].)

UHF-algebras.

AF-algebras that are inductive limits of single full matrix algebras with unit-preserving connecting mappings are called "UHF-algebras" (uniformly hyper-finite algebras) or Glimm algebras. A UHF-algebra is therefore an inductive limit of a sequence $M_{k_1}(\mathbb C)\rightarrow M_{k_2}(\mathbb C)\rightarrow\dots$, where, necessarily, each $k_j$ divides $k_{j+1}$. Setting $n_1=k_1$ and $n_j=k_j/k_{j-1}$ for $j\geq 2$, this UHF-algebra can alternatively be described as the infinite tensor product $M_{n_1}(\mathbb C)\otimes M_{n_2}(\mathbb C)\otimes\dots$. (See [a1].)

The UHF-algebra with $n_1=n_2=\dots=2$ is called the CAR-algebra; it is generated by a family of operators $\{\alpha(f):\ f\in H\}$, where $H$ is some separable infinite-dimensional Hilbert space and $\alpha$ is linear and satisfies the canonical anti-commutation relations (cf. also Commutation and anti-commutation relationships, representation of):

$\begin{align*} \alpha(f)\,\alpha(g)\,+\,\alpha(g)\,\alpha(f)&=0\\ \alpha(f)\,\alpha(g)^*\,+\,\alpha(g)^*\,\alpha(f)&=(f,g)\,1\\ \end{align*} $

(See [a7].)

-theory and classification.

By the -theory for -algebras, one can associate a triple to each -algebra . is the countable Abelian group of formal differences of equivalence classes of projections in matrix algebras over , and and are the subsets of those elements in that are represented by projections in some matrix algebra over , respectively, by projections in itself. The -group of an AF-algebra is always zero.

The classification theorem for AF-algebras says that two AF-algebras and are -isomorphic if and only if the triples and are isomorphic, i.e., if and only if there exists a group isomorphism such that and . If this is the case, then there exists an isomorphism such that . Moreover, any homomorphism such that is induced by a -homomorphism , and if are two -homomorphisms, then if and only if and are homotopic (through a continuous path of -homomorphisms from to ).

An ordered Abelian group is said to have the Riesz interpolation property if whenever with , there exists a such that . is called unperforated if , for some integer and some , implies that . The Effros–Handelman–Shen theorem says that a countable ordered Abelian group is the -theory of some AF-algebra if and only if it has the Riesz interpolation property and is unperforated. (See [a3], [a5], [a8], and [a6].)

A conjecture belonging to the Elliott classification program asserts that a -algebra is an AF-algebra if it looks like an AF-algebra! More precisely, suppose that is a separable, nuclear -algebra which has stable rank one and real rank zero, and suppose that and that is unperforated ( must necessarily have the Riesz interpolation property when is assumed to be of real rank zero). Does it follow that is an AF-algebra? This conjecture has been confirmed in some specific non-trivial cases. (See [a9].)

Traces and ideals.

The -theory of an AF-algebra not only serves as a classifying invariant, it also explicitly reveals some of the structure of the algebra, for example its traces and its ideal structure. Recall that a (positive) trace on a -algebra is a (positive) linear mapping satisfying the trace property: for all . An "ideal" means a closed two-sided ideal.

A state on an ordered Abelian group is a group homomorphism satisfying . An order ideal of is a subgroup of with the property that generates , and if , , and , then . A trace on induces a state on by

where , are projections in (or in a matrix algebra over ); and given an ideal in , the image of the induced mapping (which happens to be injective, when is an AF-algebra) is an order ideal of . For AF-algebras, the mappings and are bijections. In particular, if is simple as an ordered group, then must be simple.

If a -algebra has a unit, then the set of tracial states (i.e., positive traces that take the value on the unit) is a Choquet simplex. Using the characterizations above, one can, for each metrizable Choquet simplex, find a simple unital AF-algebra whose trace simplex is affinely homeomorphic to the given Choquet simplex. Hence, for example, simple unital -algebras can have more than one trace. (See [a3] and [a5].)

Embeddings into AF-algebras.

One particularly interesting, and still not fully investigated, application of AF-algebras is to find for a -algebra an AF-algebra and an embedding which induces an interesting (say injective) mapping . Since is positive, the positive cone of must be contained in the pre-image of . For example, the order structure of the -group of the irrational rotation -algebra was determined by embedding into an AF-algebra with (as an ordered group). As a corollary to this, it was proved that if and only if or . (See [a4].)

Along another interesting avenue there have been produced embeddings of into appropriate AF-algebras inducing injective -theory mappings. This suggests that the "cohomological dimension" of these AF-algebras should be at least .

References

[a1]	J. Glimm, "On a certain class of operator algebras" Trans. Amer. Math. Soc. , 95 (1960) pp. 318–340 MR0112057 Zbl 0094.09701
[a2]	O. Bratteli, "Inductive limits of finite-dimensional -algebras" Trans. Amer. Math. Soc. , 171 (1972) pp. 195–234 MR312282
[a3]	G.A. Elliott, "On the classification of inductive limits of sequences of semisimple finite-dimensional algebras" J. Algebra , 38 (1976) pp. 29–44 MR0397420 Zbl 0323.46063
[a4]	M. Pimsner, D. Voiculescu, "Imbedding the irrational rotation algebras into AF-algebras" J. Operator Th. , 4 (1980) pp. 201–210 MR595412
[a5]	E. Effros, D. Handelman, C.-L. Shen, "Dimension groups and their affine representations" Amer. J. Math. , 102 (1980) pp. 385–407 MR0564479 Zbl 0457.46047
[a6]	E. Effros, "Dimensions and -algebras" , CBMS Regional Conf. Ser. Math. , 46 , Amer. Math. Soc. (1981) MR0623762
[a7]	O. Bratteli, D.W. Robinson, "Operator algebras and quantum statistical mechanics" , II , Springer (1981) MR0611508 Zbl 0463.46052
[a8]	B. Blackadar, "-theory for operator algebras" , MSRI publication , 5 , Springer (1986) MR0859867 Zbl 0597.46072
[a9]	G.A. Elliott, "The classification problem for amenable -algebras" , Proc. Internat. Congress Mathem. (Zürich, 1994) , Birkhäuser (1995) pp. 922–932