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− | ''approximately finite-dimensional algebra'' | + | ''(Automatically converted into $\TeX$.)'' |
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− | 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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− | 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" />;
| + | == Approximately Finite-dimensional algebra. == |
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− | 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" />.
| + | 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]]]): |
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− | ==Bratteli diagrams.==
| + | # 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 $j=1,\dots,n$; |
− | 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" />).
| + | # 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 $A$. |
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− | 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]]].)
| + | ==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]]].) |
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| ==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]]].) | + | 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]]].) |
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− | 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]]): | + | 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]]): |
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− | <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>
| + | $$ |
| + | \alpha(f)\,\alpha(g)\,+\,\alpha(g)\,\alpha(f)=0, |
| + | $$ |
| + | $$ |
| + | \alpha(f)\,\alpha(g)^*\,+\,\alpha(g)^*\,\alpha(f)=(f,g)\,1. |
| + | $$ |
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− | <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>
| + | (See [[#References|[a7]]].) |
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− | (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. |
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− | ==<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110420/a11042047.png" />-theory and classification.==
| + | The classification theorem for AF-algebras says that two AF-algebras $ A $ |
− | 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.
| + | 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 $). |
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− | 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" />).
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− | 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]]].) | + | 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 <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]]].) | + | 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|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 |
| | | |
− | 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
| + | $$ |
| + | K _{0} ( \tau ) \left ( [ p ] _{0} - [ q ] _{0} \right ) = \tau ( p ) - \tau ( q ) , |
| + | $$ |
| | | |
− | <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>
| |
| | | |
− | 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. | + | 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 <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]]].) | + | 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==== |
− | <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> | + | <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]] |
(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]):
- 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), that every finite-dimensional$C^*$-algebra is isomorphic to the direct sum of full matrix algebras over the field of complex numbers. Property 2 says that each AF-algebra is the inductive limit of a sequence $A_1\rightarrow A_2\rightarrow\dots$ of finite-dimensional $C^*$-algebras, where the connecting mappings $A_j\rightarrow A_{j+1}$ are ${}^*$-preserving homomorphisms. If two such sequences $A_1\rightarrow A_2\rightarrow\dots$ and $B_1\rightarrow B_2\rightarrow\dots$ define isomorphic AF-algebras, then already the algebraic inductive limits of the two sequences are isomorphic (as algebras over $\mathbb C$).
All essential information of a sequence $A_1\rightarrow A_2\rightarrow\dots$ of finite-dimensional $C^*$-algebras with connecting mappings can be expressed in a so-called Bratteli diagram. The Bratteli diagram is a graph, divided into rows, whose vertices in the $j$th row correspond to the direct summands of $A_j$ isomorphic to a full matrix algebra, and where the edges between the $j$th and the $(j+1)$st row describe the connecting mapping $A_j\rightarrow A_{j+1}$. By the facts mentioned above, the construction and also the classification of AF-algebras can be reduced to a purely combinatorial problem phrased in terms of Bratteli diagrams. (See [a2].)
UHF-algebras.
AF-algebras that are inductive limits of single full matrix algebras with unit-preserving connecting mappings are called "UHF-algebras" (uniformly hyper-finite algebras) or Glimm algebras. A UHF-algebra is therefore an inductive limit of a sequence $M_{k_1}(\mathbb C)\rightarrow M_{k_2}(\mathbb C)\rightarrow\dots$, where, necessarily, each $k_j$ divides $k_{j+1}$. Setting $n_1=k_1$ and $n_j=k_j/k_{j-1}$ for $j\geq 2$, this UHF-algebra can alternatively be described as the infinite tensor product $M_{n_1}(\mathbb C)\otimes M_{n_2}(\mathbb C)\otimes\dots$. (See [a1].)
The UHF-algebra with $n_1=n_2=\dots=2$ is called the CAR-algebra; it is generated by a family of operators $\{\alpha(f):\ f\in H\}$, where $H$ is some separable infinite-dimensional Hilbert space and $\alpha$ is linear and satisfies the canonical anti-commutation relations (cf. also Commutation and anti-commutation relationships, representation of):
$$
\alpha(f)\,\alpha(g)\,+\,\alpha(g)\,\alpha(f)=0,
$$
$$
\alpha(f)\,\alpha(g)^*\,+\,\alpha(g)^*\,\alpha(f)=(f,g)\,1.
$$
(See [a7].)
$ K $-theory and classification.
By the $ K $-
theory for $ C ^{*} $-
algebras, one can associate a triple $ ( K _{0} ( A ) , K _{0} ( A ) ^{+} , \Sigma ( A ) ) $
to each $ C ^{*} $-
algebra $ A $.
$ K _{0} ( A ) $
is the countable Abelian group of formal differences of equivalence classes of projections in matrix algebras over $ A $,
and $ K _{0} ( A ) ^{+} $
and $ \Sigma ( A ) $
are the subsets of those elements in $ K _{0} ( A ) $
that are represented by projections in some matrix algebra over $ A $,
respectively, by projections in $ A $
itself. The $ K _{1} $-
group of an AF-algebra is always zero.
The classification theorem for AF-algebras says that two AF-algebras $ A $
and $ B $
are $ ^{*} $-
isomorphic if and only if the triples $ ( K _{0} ( A ) ,K _{0} ( A ) ^{+} , \Sigma ( A ) ) $
and $ ( K _{0} ( B ) , K _{0} ( B ) ^{+} , \Sigma ( B ) ) $
are isomorphic, i.e., if and only if there exists a group isomorphism $ \alpha : {K _{0} ( A )} \rightarrow {K _{0} ( B )} $
such that $ \alpha ( K _{0} ( A ) ^{+} ) = K _{0} ( B ) ^{+} $
and $ \alpha ( \Sigma ( A ) ) = \Sigma ( B ) $.
If this is the case, then there exists an isomorphism $ \varphi : A \rightarrow B $
such that $ K _{0} ( \varphi ) = \alpha $.
Moreover, any homomorphism $ \alpha : {K _{0} ( A )} \rightarrow {K _{0} ( B )} $
such that $ \alpha ( \Sigma ( A ) ) \subseteq \Sigma ( B ) $
is induced by a $ ^{*} $-
homomorphism $ \varphi : A \rightarrow B $,
and if $ {\varphi, \psi} : A \rightarrow B $
are two $ ^{*} $-
homomorphisms, then $ K _{0} ( \varphi ) = K _{0} ( \psi ) $
if and only if $ \varphi $
and $ \psi $
are homotopic (through a continuous path of $ ^{*} $-
homomorphisms from $ A $
to $ B $).
An ordered Abelian group $ ( G,G ^{+} ) $
is said to have the Riesz interpolation property if whenever $ x _{1} ,x _{2} ,y _{1} ,y _{2} \in G $
with $ x _{i} \leq y _{j} $,
there exists a $ z \in G $
such that $ x _{i} \leq z \leq y _{j} $.
$ ( G,G ^{+} ) $
is called unperforated if $ nx \geq 0 $,
for some integer $ n > 0 $
and some $ x \in G $,
implies that $ x \geq 0 $.
The Effros–Handelman–Shen theorem says that a countable ordered Abelian group $ ( G,G ^{+} ) $
is the $ K $-
theory of some AF-algebra if and only if it has the Riesz interpolation property and is unperforated. (See [a3], [a5], [a8], and [a6].)
A conjecture belonging to the Elliott classification program asserts that a $ C ^{*} $-
algebra is an AF-algebra if it looks like an AF-algebra! More precisely, suppose that $ A $
is a separable, nuclear $ C ^{*} $-
algebra which has stable rank one and real rank zero, and suppose that $ K _{1} ( A ) = 0 $
and that $ K _{0} ( A ) $
is unperforated ( $ K _{0} ( A ) $
must necessarily have the Riesz interpolation property when $ A $
is assumed to be of real rank zero). Does it follow that $ A $
is an AF-algebra? This conjecture has been confirmed in some specific non-trivial cases. (See [a9].)
Traces and ideals.
The $ K $-
theory of an AF-algebra not only serves as a classifying invariant, it also explicitly reveals some of the structure of the algebra, for example its traces and its ideal structure. Recall that a (positive) trace on a $C ^{*}$-algebra $ A $
is a (positive) linear mapping $ \tau : A \rightarrow \mathbf C $
satisfying the trace property: $ \tau ( xy ) = \tau ( yx ) $
for all $ x,y \in A $.
An "ideal" means a closed two-sided ideal.
A state $ f $
on an ordered Abelian group $ ( G,G ^{+} ) $
is a group homomorphism $ f : G \rightarrow \mathbf R $
satisfying $ f ( G ^{+} ) \subseteq \mathbf R ^{+} $.
An order ideal $ H $
of $ ( G,G ^{+} ) $
is a subgroup of $ G $
with the property that $ H ^{+} = G ^{+} \cap H $
generates $ H $,
and if $ x \in H ^{+} $,
$ y \in G ^{+} $,
and $ y \leq x $,
then $ y \in H $.
A trace $ \tau $
on $ A $
induces a state on $ ( K _{0} ( A ) ,K _{0} ( A ) ^{+} ) $
by
$$
K _{0} ( \tau ) \left ( [ p ] _{0} - [ q ] _{0} \right ) = \tau ( p ) - \tau ( q ) ,
$$
where $ p $,
$ q $
are projections in $ A $(
or in a matrix algebra over $ A $);
and given an ideal $ I $
in $ A $,
the image $ I _{*} $
of the induced mapping $ K _{0} ( I ) \rightarrow K _{0} ( A ) $(
which happens to be injective, when $ A $
is an AF-algebra) is an order ideal of $ K _{0} ( A ) $.
For AF-algebras, the mappings $ \tau \mapsto K _{0} ( \tau ) $
and $ I \mapsto I _{*} $
are bijections. In particular, if $ K _{0} ( A ) $
is simple as an ordered group, then $ A $
must be simple.
If a $ C ^{*} $-
algebra $ A $
has a unit, then the set of tracial states (i.e., positive traces that take the value $ 1 $
on the unit) is a Choquet simplex. Using the characterizations above, one can, for each metrizable Choquet simplex, find a simple unital AF-algebra whose trace simplex is affinely homeomorphic to the given Choquet simplex. Hence, for example, simple unital $ C ^{*} $-
algebras can have more than one trace. (See [a3] and [a5].)
Embeddings into AF-algebras.
One particularly interesting, and still not fully investigated, application of AF-algebras is to find for a $ C ^{*} $-
algebra $ A $
an AF-algebra $ B $
and an embedding $ \varphi : A \rightarrow B $
which induces an interesting (say injective) mapping $ {K _{0} ( \varphi )} : {K _{0} ( A )} \rightarrow {K _{0} ( B )} $.
Since $ K _{0} ( \varphi ) $
is positive, the positive cone $ K _{0} ( A ) ^{+} $
of $ K _{0} ( A ) $
must be contained in the pre-image of $ K _{0} ( B ) ^{+} $.
For example, the order structure of the $ K _{0} $-
group of the irrational rotation $ C ^{*} $-
algebra $ A _ \theta $
was determined by embedding $ A _ \theta $
into an AF-algebra $ B $
with $ K _{0} ( B ) = \mathbf Z + \theta \mathbf Z $(
as an ordered group). As a corollary to this, it was proved that $ A _ \theta \cong A _ {\theta ^ \prime} $
if and only if $ \theta = \theta ^ \prime $
or $ \theta = 1 - \theta ^ \prime $.
(See [a4].)
Along another interesting avenue there have been produced embeddings of $ C ( S ^ {2n} ) $
into appropriate AF-algebras inducing injective $ K $-
theory mappings. This suggests that the "cohomological dimension" of these AF-algebras should be at least $ 2n $.
References
[a1] | J. Glimm, "On a certain class of operator algebras" Trans. Amer. Math. Soc. , 95 (1960) pp. 318–340 MR0112057 Zbl 0094.09701 |
[a2] | O. Bratteli, "Inductive limits of finite-dimensional $C^\star$-algebras" Trans. Amer. Math. Soc. , 171 (1972) pp. 195–234 MR312282 |
[a3] | G.A. Elliott, "On the classification of inductive limits of sequences of semisimple finite-dimensional algebras" J. Algebra , 38 (1976) pp. 29–44 MR0397420 Zbl 0323.46063 |
[a4] | M. Pimsner, D. Voiculescu, "Imbedding the irrational rotation algebras into AF-algebras" J. Operator Th. , 4 (1980) pp. 201–210 MR595412 |
[a5] | E. Effros, D. Handelman, C.-L. Shen, "Dimension groups and their affine representations" Amer. J. Math. , 102 (1980) pp. 385–407 MR0564479 Zbl 0457.46047 |
[a6] | E. Effros, "Dimensions and $C^\star$-algebras" , CBMS Regional Conf. Ser. Math. , 46 , Amer. Math. Soc. (1981) MR0623762 |
[a7] | O. Bratteli, D.W. Robinson, "Operator algebras and quantum statistical mechanics" , II , Springer (1981) MR0611508 Zbl 0463.46052 |
[a8] | B. Blackadar, "$K$-theory for operator algebras" , MSRI publication , 5 , Springer (1986) MR0859867 Zbl 0597.46072 |
[a9] | G.A. Elliott, "The classification problem for amenable $C^\star$-algebras" , Proc. Internat. Congress Mathem. (Zürich, 1994) , Birkhäuser (1995) pp. 922–932 |