# Difference between revisions of "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 $2n$.