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''approximately finite-dimensional algebra''
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== Approximately Finite-dimensional algebra. ==
  
AF-algebras form a class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a1104201.png" />-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|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a1104202.png" />-algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a1104203.png" /> 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]]]):
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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]]]):
  
i) for every finite subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a1104204.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a1104205.png" /> and for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a1104206.png" /> there exists a finite-dimensional sub-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a1104207.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a1104208.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a1104209.png" /> and a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042011.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042012.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042013.png" />;
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# 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 $j=1,\dots,n$;
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# 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 $A$.
  
ii) there exists an increasing sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042014.png" /> of finite-dimensional sub-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042015.png" />-algebras of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042016.png" /> such that the union <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042017.png" /> is norm-dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042018.png" />.
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==Bratteli diagrams. ==
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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 $\mathbb C$).
  
==Bratteli diagrams.==
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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]]].)
It follows from (an analogue of) Wedderburn's theorem (cf. [[Wedderburn–Artin theorem|Wedderburn–Artin theorem]]), that every finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042019.png" />-algebra is isomorphic to the direct sum of full matrix algebras over the field of complex numbers. Property ii) says that each AF-algebra is the [[Inductive limit|inductive limit]] of a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042020.png" /> of finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042021.png" />-algebras, where the connecting mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042022.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042023.png" />-preserving homomorphisms. If two such sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042025.png" /> define isomorphic AF-algebras, then already the algebraic inductive limits of the two sequences are isomorphic (as algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042026.png" />).
 
 
 
All essential information of a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042027.png" /> of finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042028.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042029.png" />th row correspond to the direct summands of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042030.png" /> isomorphic to a full matrix algebra, and where the edges between the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042031.png" />th and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042032.png" />st row describe the connecting mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042033.png" />. 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.==
 
==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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042034.png" />, where, necessarily, each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042035.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042036.png" />. Setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042038.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042039.png" />, this UHF-algebra can alternatively be described as the infinite tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042040.png" />. (See [[#References|[a1]]].)
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042041.png" /> is called the CAR-algebra; it is generated by a family of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042042.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042043.png" /> is some separable infinite-dimensional [[Hilbert space|Hilbert space]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042044.png" /> 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]]):
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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]]):
  
<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/a11042045.png" /></td> </tr></table>
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$\begin{align*}
 
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\alpha(f)\,\alpha(g)\,+\,\alpha(g)\,\alpha(f)&=0\\
<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/a11042046.png" /></td> </tr></table>
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\alpha(f)\,\alpha(g)^*\,+\,\alpha(g)^*\,\alpha(f)&=(f,g)\,1\\
 +
\end{align*}
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$
  
 
(See [[#References|[a7]]].)
 
(See [[#References|[a7]]].)
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==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]].
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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]].
  
 
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 <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
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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 <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]]].)
  
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" />.
+
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====
<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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> O. Bratteli,   "Inductive limits of finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420167.png" />-algebras" ''Trans. Amer. Math. Soc.'' , '''171''' (1972) pp. 195–234</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</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</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</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> E. Effros,   "Dimensions and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420168.png" />-algebras" , ''CBMS Regional Conf. Ser. Math.'' , '''46''' , Amer. Math. Soc. (1981)</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)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> B. Blackadar,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420169.png" />-theory for operator algebras" , ''MSRI publication'' , '''5''' , Springer (1986)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> G.A. Elliott,   "The classification problem for amenable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420170.png" />-algebras" , ''Proc. Internat. Congress Mathem. (Zürich, 1994)'' , Birkhäuser (1995) pp. 922–932</TD></TR></table>
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<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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420167.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420168.png" />-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, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420169.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a110420170.png" />-algebras" , ''Proc. Internat. Congress Mathem. (Zürich, 1994)'' , Birkhäuser (1995) pp. 922–932</TD></TR></table>
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[[Category:Associative rings and algebras]]

Revision as of 19:49, 7 November 2014

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
How to Cite This Entry:
AF-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=AF-algebra&oldid=11299
This article was adapted from an original article by M. Rørdam (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article