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== Approximately Finite-dimensional algebra. ==
 
== Approximately Finite-dimensional algebra. ==
  
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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]]):
 
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]]):
  
$\begin{align*}
+
$$
\alpha(f)\,\alpha(g)\,+\,\alpha(g)\,\alpha(f)&=0\\
+
\alpha(f)\,\alpha(g)\,+\,\alpha(g)\,\alpha(f)=0,
\alpha(f)\,\alpha(g)^*\,+\,\alpha(g)^*\,\alpha(f)&=(f,g)\,1\\
+
$$
\end{align*}
+
$$
$
+
\alpha(f)\,\alpha(g)^*\,+\,\alpha(g)^*\,\alpha(f)=(f,g)\,1.
 +
$$
  
 
(See [[#References|[a7]]].)
 
(See [[#References|[a7]]].)
  
==<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042047.png" />-theory and classification.==
+
== $  K $-theory and classification.==
By the [[K-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042048.png" />-theory]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042049.png" />-algebras, one can associate a triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042050.png" /> to each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042051.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042052.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042053.png" /> is the countable Abelian group of formal differences of equivalence classes of projections in matrix algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042054.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042056.png" /> are the subsets of those elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042057.png" /> that are represented by projections in some matrix algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042058.png" />, respectively, by projections in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042059.png" /> itself. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042060.png" />-group of an AF-algebra is always zero.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042062.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042063.png" />-isomorphic if and only if the triples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042065.png" /> are isomorphic, i.e., if and only if there exists a group isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042066.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042067.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042068.png" />. If this is the case, then there exists an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042069.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042070.png" />. Moreover, any homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042071.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042072.png" /> is induced by a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042073.png" />-homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042074.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042075.png" /> are two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042076.png" />-homomorphisms, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042077.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042078.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042079.png" /> are homotopic (through a continuous path of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042080.png" />-homomorphisms from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042081.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042082.png" />).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042083.png" /> is said to have the Riesz interpolation property if whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042084.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042085.png" />, there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042086.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042087.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042088.png" /> is called unperforated if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042089.png" />, for some integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042090.png" /> and some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042091.png" />, implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042092.png" />. The Effros–Handelman–Shen theorem says that a countable ordered Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042093.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042094.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042095.png" />-algebra is an AF-algebra if it looks like an AF-algebra! More precisely, suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042096.png" /> is a separable, nuclear <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042097.png" />-algebra which has stable rank one and real rank zero, and suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042098.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042099.png" /> is unperforated (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420100.png" /> must necessarily have the Riesz interpolation property when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420101.png" /> is assumed to be of real rank zero). Does it follow that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420102.png" /> is an AF-algebra? This conjecture has been confirmed in some specific non-trivial cases. (See [[#References|[a9]]].)
+
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.==
 
==Traces and ideals.==
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420103.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420105.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420106.png" /> is a (positive) linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420107.png" /> satisfying the trace property: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420108.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420109.png" />. An "ideal" means a closed two-sided [[Ideal|ideal]].
+
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|ideal]].
  
A state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420110.png" /> on an ordered Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420111.png" /> is a group [[Homomorphism|homomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420112.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420113.png" />. An order ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420114.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420115.png" /> is a [[Subgroup|subgroup]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420116.png" /> with the property that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420117.png" /> generates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420118.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420119.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420120.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420121.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420122.png" />. A trace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420123.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420124.png" /> induces a state on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420125.png" /> by
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420126.png" /></td> </tr></table>
+
$$
 +
K _{0} ( \tau ) \left ( [ p ] _{0} - [ q ] _{0} \right ) = \tau ( p ) - \tau ( q ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420127.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420128.png" /> are projections in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420129.png" /> (or in a matrix algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420130.png" />); and given an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420131.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420132.png" />, the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420133.png" /> of the induced mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420134.png" /> (which happens to be injective, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420135.png" /> is an AF-algebra) is an order ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420136.png" />. For AF-algebras, the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420137.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420138.png" /> are bijections. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420139.png" /> is simple as an ordered group, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420140.png" /> must be simple.
 
  
If a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420141.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420142.png" /> has a unit, then the set of tracial states (i.e., positive traces that take the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420143.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420144.png" />-algebras can have more than one trace. (See [[#References|[a3]]] and [[#References|[a5]]].)
+
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.==
 
==Embeddings into AF-algebras.==
One particularly interesting, and still not fully investigated, application of AF-algebras is to find for a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420145.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420146.png" /> an AF-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420147.png" /> and an embedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420148.png" /> which induces an interesting (say injective) mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420149.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420150.png" /> is positive, the positive cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420151.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420152.png" /> must be contained in the pre-image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420153.png" />. For example, the order structure of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420154.png" />-group of the irrational rotation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420156.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420157.png" /> was determined by embedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420158.png" /> into an AF-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420159.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420160.png" /> (as an ordered group). As a corollary to this, it was proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420161.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420162.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420163.png" />. (See [[#References|[a4]]].)
+
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 $.
  
Along another interesting avenue there have been produced embeddings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420164.png" /> into appropriate AF-algebras inducing injective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420165.png" />-theory mappings. This suggests that the "cohomological dimension" of these AF-algebras should be at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420166.png" />.
 
  
 
====References====
 
====References====

Latest revision as of 14:30, 24 January 2020

(Automatically converted into $\TeX$.)

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 -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
How to Cite This Entry:
AF-algebra. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=AF-algebra&oldid=44337
This article was adapted from an original article by M. Rørdam (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article