AF-algebra

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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):

$$ \alpha(f)\,\alpha(g)\,+\,\alpha(g)\,\alpha(f)=0, $$ $$ \alpha(f)\,\alpha(g)^*\,+\,\alpha(g)^*\,\alpha(f)=(f,g)\,1. $$

(See [a7].)

$ K $-theory and classification.

By the $ K $- theory for $ C ^{*} $- algebras, one can associate a triple $ ( K _{0} ( A ) , K _{0} ( A ) ^{+} , \Sigma ( A ) ) $ to each $ C ^{*} $- algebra $ A $. $ K _{0} ( A ) $ is the countable Abelian group of formal differences of equivalence classes of projections in matrix algebras over $ A $, and $ K _{0} ( A ) ^{+} $ and $ \Sigma ( A ) $ are the subsets of those elements in $ K _{0} ( A ) $ that are represented by projections in some matrix algebra over $ A $, respectively, by projections in $ A $ itself. The $ K _{1} $- group of an AF-algebra is always zero.

The classification theorem for AF-algebras says that two AF-algebras $ A $ and $ B $ are $ ^{*} $- isomorphic if and only if the triples $ ( K _{0} ( A ) ,K _{0} ( A ) ^{+} , \Sigma ( A ) ) $ and $ ( K _{0} ( B ) , K _{0} ( B ) ^{+} , \Sigma ( B ) ) $ are isomorphic, i.e., if and only if there exists a group isomorphism $ \alpha : {K _{0} ( A )} \rightarrow {K _{0} ( B )} $ such that $ \alpha ( K _{0} ( A ) ^{+} ) = K _{0} ( B ) ^{+} $ and $ \alpha ( \Sigma ( A ) ) = \Sigma ( B ) $. If this is the case, then there exists an isomorphism $ \varphi : A \rightarrow B $ such that $ K _{0} ( \varphi ) = \alpha $. Moreover, any homomorphism $ \alpha : {K _{0} ( A )} \rightarrow {K _{0} ( B )} $ such that $ \alpha ( \Sigma ( A ) ) \subseteq \Sigma ( B ) $ is induced by a $ ^{*} $- homomorphism $ \varphi : A \rightarrow B $, and if $ {\varphi, \psi} : A \rightarrow B $ are two $ ^{*} $- homomorphisms, then $ K _{0} ( \varphi ) = K _{0} ( \psi ) $ if and only if $ \varphi $ and $ \psi $ are homotopic (through a continuous path of $ ^{*} $- homomorphisms from $ A $ to $ B $).

An ordered Abelian group $ ( G,G ^{+} ) $ is said to have the Riesz interpolation property if whenever $ x _{1} ,x _{2} ,y _{1} ,y _{2} \in G $ with $ x _{i} \leq y _{j} $, there exists a $ z \in G $ such that $ x _{i} \leq z \leq y _{j} $. $ ( G,G ^{+} ) $ is called unperforated if $ nx \geq 0 $, for some integer $ n > 0 $ and some $ x \in G $, implies that $ x \geq 0 $. The Effros–Handelman–Shen theorem says that a countable ordered Abelian group $ ( G,G ^{+} ) $ is the $ K $- 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 $ C ^{*} $- algebra is an AF-algebra if it looks like an AF-algebra! More precisely, suppose that $ A $ is a separable, nuclear $ C ^{*} $- algebra which has stable rank one and real rank zero, and suppose that $ K _{1} ( A ) = 0 $ and that $ K _{0} ( A ) $ is unperforated ( $ K _{0} ( A ) $ must necessarily have the Riesz interpolation property when $ A $ is assumed to be of real rank zero). Does it follow that $ A $ is an AF-algebra? This conjecture has been confirmed in some specific non-trivial cases. (See [a9].)

Traces and ideals.

The $ K $- 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 $C ^{*}$-algebra $ A $ is a (positive) linear mapping $ \tau : A \rightarrow \mathbf C $ satisfying the trace property: $ \tau ( xy ) = \tau ( yx ) $ for all $ x,y \in A $. An "ideal" means a closed two-sided ideal.

A state $ f $ on an ordered Abelian group $ ( G,G ^{+} ) $ is a group homomorphism $ f : G \rightarrow \mathbf R $ satisfying $ f ( G ^{+} ) \subseteq \mathbf R ^{+} $. An order ideal $ H $ of $ ( G,G ^{+} ) $ is a subgroup of $ G $ with the property that $ H ^{+} = G ^{+} \cap H $ generates $ H $, and if $ x \in H ^{+} $, $ y \in G ^{+} $, and $ y \leq x $, then $ y \in H $. A trace $ \tau $ on $ A $ induces a state on $ ( K _{0} ( A ) ,K _{0} ( A ) ^{+} ) $ by

$$ K _{0} ( \tau ) \left ( [ p ] _{0} - [ q ] _{0} \right ) = \tau ( p ) - \tau ( q ) , $$

where $ p $, $ q $ are projections in $ A $( or in a matrix algebra over $ A $); and given an ideal $ I $ in $ A $, the image $ I _{*} $ of the induced mapping $ K _{0} ( I ) \rightarrow K _{0} ( A ) $( which happens to be injective, when $ A $ is an AF-algebra) is an order ideal of $ K _{0} ( A ) $. For AF-algebras, the mappings $ \tau \mapsto K _{0} ( \tau ) $ and $ I \mapsto I _{*} $ are bijections. In particular, if $ K _{0} ( A ) $ is simple as an ordered group, then $ A $ must be simple.

If a $ C ^{*} $- algebra $ A $ has a unit, then the set of tracial states (i.e., positive traces that take the value $ 1 $ 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 $ C ^{*} $- 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 $ C ^{*} $- algebra $ A $ an AF-algebra $ B $ and an embedding $ \varphi : A \rightarrow B $ which induces an interesting (say injective) mapping $ {K _{0} ( \varphi )} : {K _{0} ( A )} \rightarrow {K _{0} ( B )} $. Since $ K _{0} ( \varphi ) $ is positive, the positive cone $ K _{0} ( A ) ^{+} $ of $ K _{0} ( A ) $ must be contained in the pre-image of $ K _{0} ( B ) ^{+} $. For example, the order structure of the $ K _{0} $- group of the irrational rotation $ C ^{*} $- algebra $ A _ \theta $ was determined by embedding $ A _ \theta $ into an AF-algebra $ B $ with $ K _{0} ( B ) = \mathbf Z + \theta \mathbf Z $( as an ordered group). As a corollary to this, it was proved that $ A _ \theta \cong A _ {\theta ^ \prime} $ if and only if $ \theta = \theta ^ \prime $ or $ \theta = 1 - \theta ^ \prime $. (See [a4].)

Along another interesting avenue there have been produced embeddings of $ C ( S ^ {2n} ) $ into appropriate AF-algebras inducing injective $ K $- theory mappings. This suggests that the "cohomological dimension" of these AF-algebras should be at least $ 2n $.

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 $C^\star$-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 $C^\star$-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, "$K$-theory for operator algebras" , MSRI publication , 5 , Springer (1986) MR0859867 Zbl 0597.46072
[a9]	G.A. Elliott, "The classification problem for amenable $C^\star$-algebras" , Proc. Internat. Congress Mathem. (Zürich, 1994) , Birkhäuser (1995) pp. 922–932