View source for AF-algebra

''(Automatically converted into $\TeX$.)''

{{TEX|done}}

== 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|$C^*$-algebra]] $A$ is said to be an ''AF-algebra'' if one of the following two (not obviously) equivalent conditions is satisfied (see [[#References|[a1]]], [[#References|[a2]]] or [[#References|[a6]]]):

# 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 $j=1,\dots,n$;
# 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|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|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 [[#References|[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 [[#References|[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|Hilbert space]] and $\alpha$ is linear and satisfies the canonical anti-commutation relations (cf. also [[Commutation and anti-commutation relationships, representation of|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 [[#References|[a7]]].)

== $  K $-theory and classification.==
By the [[K-theory| $  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 [[#References|[a3]]], [[#References|[a5]]], [[#References|[a8]]], and [[#References|[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 [[#References|[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|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|ideal]].

A state  $  f $
on an ordered Abelian group  $  ( G,G ^{+} ) $
is a group [[Homomorphism|homomorphism]]  $  f : G \rightarrow \mathbf R $
satisfying  $  f ( G ^{+} ) \subseteq \mathbf R ^{+} $. 
An order ideal  $  H $
of  $  ( G,G ^{+} ) $
is a [[Subgroup|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|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 [[#References|[a3]]] and [[#References|[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 [[#References|[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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Glimm, "On a certain class of operator algebras" ''Trans. Amer. Math. Soc.'' , '''95''' (1960) pp. 318–340 {{MR|0112057}} {{ZBL|0094.09701}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> O. Bratteli, "Inductive limits of finite-dimensional $C^\star$-algebras" ''Trans. Amer. Math. Soc.'' , '''171''' (1972) pp. 195–234 {{MR|312282}} {{ZBL|}} </TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top"> G.A. Elliott, "On the classification of inductive limits of sequences of semisimple finite-dimensional algebras" ''J. Algebra'' , '''38''' (1976) pp. 29–44 {{MR|0397420}} {{ZBL|0323.46063}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> M. Pimsner, D. Voiculescu, "Imbedding the irrational rotation algebras into AF-algebras" ''J. Operator Th.'' , '''4''' (1980) pp. 201–210 {{MR|595412}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> E. Effros, D. Handelman, C.-L. Shen, "Dimension groups and their affine representations" ''Amer. J. Math.'' , '''102''' (1980) pp. 385–407 {{MR|0564479}} {{ZBL|0457.46047}} </TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top"> E. Effros, "Dimensions and $C^\star$-algebras" , ''CBMS Regional Conf. Ser. Math.'' , '''46''' , Amer. Math. Soc. (1981) {{MR|0623762}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> O. Bratteli, D.W. Robinson, "Operator algebras and quantum statistical mechanics" , '''II''' , Springer (1981) {{MR|0611508}} {{ZBL|0463.46052}} </TD></TR>
<TR><TD valign="top">[a8]</TD> <TD valign="top"> B. Blackadar, "$K$-theory for operator algebras" , ''MSRI publication'' , '''5''' , Springer (1986) {{MR|0859867}} {{ZBL|0597.46072}} </TD></TR>
<TR><TD valign="top">[a9]</TD> <TD valign="top"> G.A. Elliott, "The classification problem for amenable $C^\star$-algebras" , ''Proc. Internat. Congress Mathem. (Zürich, 1994)'' , Birkhäuser (1995) pp. 922–932</TD></TR>
</table>

[[Category:Associative rings and algebras]]