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''abstract von Neumann algebra''
 
''abstract von Neumann algebra''
  
An algebra from a strictly larger class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a1203102.png" />-algebras than the class of von Neumann algebras (cf. also [[Von Neumann algebra|von Neumann algebra]]). Such algebras were introduced by I. Kaplansky [[#References|[a9]]], [[#References|[a10]]], [[#References|[a11]]], [[#References|[a12]]], originally as a means of abstracting the algebraic properties of von Neumann algebras from their topological properties. Since von Neumann algebras are also known as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a1203104.png" />-algebras, such algebras were termed abstract <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a1203106.png" />-algebras, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a1203107.png" />-algebras. Indeed the  "classical"  approach to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a1203108.png" />-algebras was devoted to showing how closely their behaviour corresponded to that of von Neumann algebras. See [[#References|[a1]]], and its extensive references, for a scholarly exposition of this classical material. However, in recent years, much effort has been devoted to investigating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a1203109.png" />-algebras whose properties can be markedly different from their von Neumann cousins.
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An algebra from a strictly larger class of $C ^ { * }$-algebras than the class of von Neumann algebras (cf. also [[Von Neumann algebra|von Neumann algebra]]). Such algebras were introduced by I. Kaplansky [[#References|[a9]]], [[#References|[a10]]], [[#References|[a11]]], [[#References|[a12]]], originally as a means of abstracting the algebraic properties of von Neumann algebras from their topological properties. Since von Neumann algebras are also known as $W ^ { * }$-algebras, such algebras were termed abstract $W ^ { * }$-algebras, or $A W ^ { * }$-algebras. Indeed the  "classical"  approach to $A W ^ { * }$-algebras was devoted to showing how closely their behaviour corresponded to that of von Neumann algebras. See [[#References|[a1]]], and its extensive references, for a scholarly exposition of this classical material. However, in recent years, much effort has been devoted to investigating $A W ^ { * }$-algebras whose properties can be markedly different from their von Neumann cousins.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031010.png" /> be a [[C*-algebra|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031011.png" />-algebra]] with a unit element. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031012.png" /> be the set of self-adjoint elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031013.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031014.png" /> has a natural partial ordering which organizes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031015.png" /> as a partially ordered real vector space with order-unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031016.png" /> (cf. also [[Semi-ordered space|Semi-ordered space]]). The positive cone of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031017.png" /> for this partial ordering is the set of all elements of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031018.png" />. When each upper bounded, upward-directed subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031019.png" /> has a least upper bound, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031020.png" /> is said to be monotone complete. All von Neumann algebras are monotone complete but the converse is false. To see this, it suffices to give examples of commutative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031022.png" />-algebras which are monotone complete but which are not von Neumann algebras.
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Let $M$ be a [[C*-algebra|$C ^ { * }$-algebra]] with a unit element. Let $M _ { \operatorname{sa} }$ be the set of self-adjoint elements of $M$. Then $M _ { \operatorname{sa} }$ has a natural partial ordering which organizes $M _ { \operatorname{sa} }$ as a partially ordered real vector space with order-unit $1$ (cf. also [[Semi-ordered space|Semi-ordered space]]). The positive cone of $M _ { \operatorname{sa} }$ for this partial ordering is the set of all elements of the form $zz ^ { * }$. When each upper bounded, upward-directed subset of $M _ { \operatorname{sa} }$ has a least upper bound, then $M$ is said to be monotone complete. All von Neumann algebras are monotone complete but the converse is false. To see this, it suffices to give examples of commutative $C ^ { * }$-algebras which are monotone complete but which are not von Neumann algebras.
  
An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031024.png" />-algebra is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031025.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031026.png" />, with a unit, such that each maximal commutative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031027.png" />-subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031028.png" /> is monotone complete. Clearly each monotone complete <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031029.png" />-algebra is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031030.png" />-algebra and every commutative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031031.png" />-algebra is monotone complete. It is natural to ask if every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031032.png" />-algebra is monotone complete. Despite the important advances of [[#References|[a3]]] this question is not yet (1999) settled.
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An $A W ^ { * }$-algebra is a $C ^ { * }$-algebra $A$, with a unit, such that each maximal commutative $*$-subalgebra of $A$ is monotone complete. Clearly each monotone complete $C ^ { * }$-algebra is an $A W ^ { * }$-algebra and every commutative $A W ^ { * }$-algebra is monotone complete. It is natural to ask if every $A W ^ { * }$-algebra is monotone complete. Despite the important advances of [[#References|[a3]]] this question is not yet (1999) settled.
  
Each commutative unital <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031033.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031034.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031035.png" />-isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031036.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031037.png" />-algebra of all complex-valued continuous functions on a compact [[Hausdorff space|Hausdorff space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031038.png" />. Then the [[Commutative algebra|commutative algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031039.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031040.png" />-algebra precisely when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031041.png" /> is extremally disconnected, that is, the closure of each open subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031042.png" /> is open (cf. also [[Extremally-disconnected space|Extremally-disconnected space]]). It follows from the Stone representation theorem for Boolean algebras (cf. also [[Boolean algebra|Boolean algebra]]) that the projections in a commutative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031043.png" />-algebra form a complete Boolean algebra and, conversely, all complete Boolean algebras arise in this way.
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Each commutative unital $C ^ { * }$-algebra $A$ is $*$-isomorphic to $C ( E )$, the $*$-algebra of all complex-valued continuous functions on a compact [[Hausdorff space|Hausdorff space]] $E$. Then the [[Commutative algebra|commutative algebra]] $A$ is an $A W ^ { * }$-algebra precisely when $E$ is extremally disconnected, that is, the closure of each open subset of $E$ is open (cf. also [[Extremally-disconnected space|Extremally-disconnected space]]). It follows from the Stone representation theorem for Boolean algebras (cf. also [[Boolean algebra|Boolean algebra]]) that the projections in a commutative $A W ^ { * }$-algebra form a complete Boolean algebra and, conversely, all complete Boolean algebras arise in this way.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031044.png" /> be a [[Topological space|topological space]] which is homeomorphic to a complete separable [[Metric space|metric space]] with no isolated points; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031045.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031046.png" />-algebra of all bounded, Borel measurable, complex-valued functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031047.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031048.png" /> be the ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031049.png" /> consisting of all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031050.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031051.png" /> is meagre, that is, of first Baire category (cf. also [[Baire classes|Baire classes]]). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031052.png" /> is a commutative monotone complete <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031053.png" />-algebra which is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031054.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031055.png" /> is a compact extremally disconnected space. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031056.png" />, which is independent of the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031057.png" />, is known as the Dixmier algebra. It can be shown that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031058.png" /> has no states which are normal. It follows from this that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031059.png" /> is not a [[Von Neumann algebra|von Neumann algebra]].
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Let $K$ be a [[Topological space|topological space]] which is homeomorphic to a complete separable [[Metric space|metric space]] with no isolated points; let $B ( K )$ be the $*$-algebra of all bounded, Borel measurable, complex-valued functions on $K$. Let $M ( K )$ be the ideal of $B ( K )$ consisting of all functions $f$ for which $\{ x \in X : f ( x ) \neq 0 \}$ is meagre, that is, of first Baire category (cf. also [[Baire classes|Baire classes]]). Then $B ( K ) / M ( K )$ is a commutative monotone complete $C ^ { * }$-algebra which is isomorphic to $C ( S )$, where $S$ is a compact extremally disconnected space. The algebra $C ( S )$, which is independent of the choice of $K$, is known as the Dixmier algebra. It can be shown that $C ( S )$ has no states which are normal. It follows from this that $C ( S )$ is not a [[Von Neumann algebra|von Neumann algebra]].
  
The classification of von Neumann algebras into Type I, Type-II, and Type-III (cf. also [[Von Neumann algebra|von Neumann algebra]]) can be extended to give a similar classification for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031063.png" />-algebras. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031064.png" /> be a Type-I <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031065.png" />-algebra and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031066.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031067.png" />-algebra embedded as a subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031068.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031069.png" /> contains the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031070.png" /> and if the lattice of projections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031071.png" /> is a complete sublattice of the lattice of projections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031072.png" />, then K. Saitô [[#References|[a16]]] proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031073.png" /> equals its bi-commutant in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031074.png" />. This result extends earlier results by J. Feldman and by H. Widom and builds on the elegant characterization by G.K. Pedersen of von Neumann algebras [[#References|[a14]]], [[#References|[a15]]]. See also [[#References|[a5]]]. By contrast, M. Ozawa [[#References|[a13]]] showed that Type-I <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031075.png" />-algebras can exhibit pathological properties.
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The classification of von Neumann algebras into Type I, Type-II, and Type-III (cf. also [[Von Neumann algebra|von Neumann algebra]]) can be extended to give a similar classification for $A W ^ { * }$-algebras. Let $B$ be a Type-I $A W ^ { * }$-algebra and let $A$ be an $A W ^ { * }$-algebra embedded as a subalgebra of $B$. If $A$ contains the centre of $B$ and if the lattice of projections of $A$ is a complete sublattice of the lattice of projections of $B$, then K. Saitô [[#References|[a16]]] proved that $A$ equals its bi-commutant in $B$. This result extends earlier results by J. Feldman and by H. Widom and builds on the elegant characterization by G.K. Pedersen of von Neumann algebras [[#References|[a14]]], [[#References|[a15]]]. See also [[#References|[a5]]]. By contrast, M. Ozawa [[#References|[a13]]] showed that Type-I $A W ^ { * }$-algebras can exhibit pathological properties.
  
An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031076.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031077.png" /> is said to be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031079.png" />-factor if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031080.png" /> has trivial centre, that is,
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An $A W ^ { * }$-algebra $A$ is said to be an $A W ^ { * }$-factor if $A$ has trivial centre, that is,
  
<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/a120/a120310/a12031081.png" /></td> </tr></table>
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\begin{equation*} \{ z \in A : z a = a z \;\text { for each } a \in A \} \end{equation*}
  
is one-dimensional. An early result of I. Kaplansky showed that each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031082.png" />-factor of Type I was, in fact, a von Neumann algebra. This made it reasonable for him to ask if the same were true for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031083.png" />-factors of Type II and Type III. For Type II there are partial results, described below, which make it plausible to conjecture that all Type-II <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031084.png" />-factors are von Neumann algebras. If this could be established then this would have important implications for separable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031085.png" />-algebras [[#References|[a2]]], [[#References|[a7]]]. For Type-III <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031086.png" />-factors the situation is completely different. Examples of such factors which are not von Neumann algebras are described below.
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is one-dimensional. An early result of I. Kaplansky showed that each $A W ^ { * }$-factor of Type I was, in fact, a von Neumann algebra. This made it reasonable for him to ask if the same were true for $A W ^ { * }$-factors of Type II and Type III. For Type II there are partial results, described below, which make it plausible to conjecture that all Type-II $A W ^ { * }$-factors are von Neumann algebras. If this could be established then this would have important implications for separable $C ^ { * }$-algebras [[#References|[a2]]], [[#References|[a7]]]. For Type-III $A W ^ { * }$-factors the situation is completely different. Examples of such factors which are not von Neumann algebras are described below.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031087.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031088.png" />-factor of Type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031089.png" />. Then it was shown in [[#References|[a21]]] that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031090.png" /> possesses a faithful state, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031092.png" /> possesses a faithful normal state and hence is a von Neumann factor of Type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031093.png" />. It follows from this that when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031094.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031095.png" />-factor of Type II which possesses a faithful state, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031096.png" /> is a von Neumann algebra [[#References|[a5]]], [[#References|[a20]]]. By contrast, there exist monotone complete <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031097.png" />-factors of Type III which possess faithful states but which are not von Neumann algebras.
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Let $A$ be an $A W ^ { * }$-factor of Type $\operatorname{II} _ { 1 }$. Then it was shown in [[#References|[a21]]] that if $A$ possesses a faithful state, then $A$ possesses a faithful normal state and hence is a von Neumann factor of Type $\operatorname{II} _ { 1 }$. It follows from this that when $B$ is an $A W ^ { * }$-factor of Type II which possesses a faithful state, then $B$ is a von Neumann algebra [[#References|[a5]]], [[#References|[a20]]]. By contrast, there exist monotone complete $A W ^ { * }$-factors of Type III which possess faithful states but which are not von Neumann algebras.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031098.png" /> be a countable [[Group|group]] of homeomorphisms of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a12031099.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310100.png" /> is homeomorphic to a complete separable metric space with no isolated points. Let the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310101.png" /> be free and let there exist a dense <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310102.png" />-orbit. The action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310103.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310104.png" /> induces a free, generically ergodic action (of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310105.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310106.png" />, the Dixmier algebra). Then there exists a corresponding cross product algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310107.png" /> which is a monotone complete <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310108.png" />-factor of Type III. Since this algebra contains a maximal commutative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310109.png" />-subalgebra isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310110.png" />, which is not a von Neumann algebra, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310111.png" /> is not a von Neumann algebra. The first examples of Type-III factors which were not von Neumann algebras were constructed, independently, by O. Takenouchi and J. Dyer. Their respective examples were of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310112.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310113.png" /> for (different) Abelian groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310114.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310115.png" />, see [[#References|[a18]]]. As a corollary of the Sullivan–Weiss–Wright theorem [[#References|[a19]]], the Takenouchi and Dyer factors are isomorphic. Much more is true. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310116.png" /> is independent of the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310117.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310118.png" /> provided the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310119.png" /> is free and generically ergodic. For example, if one takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310120.png" /> to be the additive group of integers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310121.png" /> to be the free group on two generators the corresponding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310122.png" />-factors are isomorphic. This is surprisingly different from the situation for von Neumann algebras. For a particularly lucid account of monotone cross-products see [[#References|[a18]]].
+
Let $G$ be a countable [[Group|group]] of homeomorphisms of a topological space $K$, where $K$ is homeomorphic to a complete separable metric space with no isolated points. Let the action of $G$ be free and let there exist a dense $G$-orbit. The action of $G$ on $K$ induces a free, generically ergodic action (of $G$ on $B ( K ) / M ( K ) = C ( S )$, the Dixmier algebra). Then there exists a corresponding cross product algebra $M ( C ( S ) , \alpha , G )$ which is a monotone complete $A W ^ { * }$-factor of Type III. Since this algebra contains a maximal commutative $*$-subalgebra isomorphic to $C ( S )$, which is not a von Neumann algebra, $M ( C ( S ) , \alpha , G )$ is not a von Neumann algebra. The first examples of Type-III factors which were not von Neumann algebras were constructed, independently, by O. Takenouchi and J. Dyer. Their respective examples were of the form $M ( C ( S ) , \alpha _ { 1 } , G _ { 1 } )$ and $M ( C ( S ) , \alpha _ { 2 } , G _ { 2 } )$ for (different) Abelian groups $G_1$ and $G_2$, see [[#References|[a18]]]. As a corollary of the Sullivan–Weiss–Wright theorem [[#References|[a19]]], the Takenouchi and Dyer factors are isomorphic. Much more is true. The algebra $M ( C ( S ) , \alpha , G )$ is independent of the choice of $G$ and $\alpha$ provided the action of $G$ is free and generically ergodic. For example, if one takes $G_1$ to be the additive group of integers and $G_2$ to be the free group on two generators the corresponding $A W ^ { * }$-factors are isomorphic. This is surprisingly different from the situation for von Neumann algebras. For a particularly lucid account of monotone cross-products see [[#References|[a18]]].
  
Another approach to constructing monotone complete Type-III <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310123.png" />-factors which are not von Neumann algebras goes as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310124.png" /> be a unital <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310125.png" />-algebra, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310126.png" /> be the Pedersen–Borel-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310128.png" /> envelope of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310129.png" /> on the universal representation space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310130.png" /> [[#References|[a15]]]. Then there is a  "meagre"  ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310131.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310132.png" /> such that the quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310133.png" /> is a monotone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310134.png" />-complete <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310135.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310136.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310137.png" /> is embedded as an order-dense subalgebra. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310138.png" /> is separable, simple and infinite dimensional, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310139.png" /> is a monotone complete <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310140.png" />-factor of Type III which is never a von Neumann algebra [[#References|[a22]]], [[#References|[a23]]]. This type of completion has been extensively generalized by M. Hamana [[#References|[a6]]].
+
Another approach to constructing monotone complete Type-III $A W ^ { * }$-factors which are not von Neumann algebras goes as follows. Let $A$ be a unital $C ^ { * }$-algebra, let $A ^ { \infty }$ be the Pedersen–Borel-$*$ envelope of $A$ on the universal representation space of $A$ [[#References|[a15]]]. Then there is a  "meagre"  ideal $M$ in $A ^ { \infty }$ such that the quotient $A ^ { \infty } / M$ is a monotone $\sigma$-complete $C ^ { * }$-algebra $\hat{A}$ in which $A$ is embedded as an order-dense subalgebra. When $A$ is separable, simple and infinite dimensional, then $\hat{A}$ is a monotone complete $A W ^ { * }$-factor of Type III which is never a von Neumann algebra [[#References|[a22]]], [[#References|[a23]]]. This type of completion has been extensively generalized by M. Hamana [[#References|[a6]]].
  
Although much progress has been made in understanding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310141.png" />-factors, many unsolved problems remain.
+
Although much progress has been made in understanding $A W ^ { * }$-factors, many unsolved problems remain.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.K. Berberian,  "Baer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310142.png" />-rings" , Springer  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Blackadar,  D. Handelman,  "Dimension functions and traces on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310143.png" />-algebras"  ''J. Funct. Anal.'' , '''45'''  (1982)  pp. 297–340</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Christensen,  G.K. Pedersen,  "Properly infinite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310144.png" />-algebras are monotone sequentially complete"  ''Bull. London Math. Soc.'' , '''16'''  (1984)  pp. 407–410</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Dixmier,  "Sur certains espace considérés par M.H. Stone"  ''Summa Brasil. Math.'' , '''2'''  (1951)  pp. 151–182</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  G.A. Elliott,  K. Saitô,  J.D.M. Wright,  "Embedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310145.png" />-algebras as double commutants in Type I algebras"  ''J. London Math. Soc.'' , '''28'''  (1983)  pp. 376–384</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M. Hamana,  "Regular embeddings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310146.png" />-algebras in monotone complete <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310147.png" />-algebras"  ''J. Math. Soc. Japan'' , '''33'''  (1981)  pp. 159–183</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  D. Handelman,  "Homomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310148.png" />-algebras to finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310149.png" />-algebras"  ''Michigan Math. J.'' , '''28'''  (1981)  pp. 229–240</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  R.V. Kadison,  G.K. Pedersen,  "Equivalence in operator algebras"  ''Math. Scand.'' , '''27'''  (1970)  pp. 205–222</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  I. Kaplansky,  "Projections in Banach algebras"  ''Ann. Math.'' , '''53'''  (1951)  pp. 235–249</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  I. Kaplansky,  "Algebras of Type I" ''Ann. Math.'' , '''56''' (1952) pp. 460–472</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  I. Kaplansky,  "Modules over operator algebras"  ''Amer. J. Math.'' , '''75'''  (1953)  pp. 839–858</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  I. Kaplansky,  "Rings of operators" , Benjamin (1968)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  M. Ozawa,  "Nonuniqueness of the cardinality attached to homogeneous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310150.png" />-algebras"  ''Proc. Amer. Math. Soc.'' , '''93'''  (1985)  pp. 681–684</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  G.K. Pedersen,  "Operator algebras with weakly closed Abelian subalgebras"  ''Bull. London Math. Soc.'' , '''4'''  (1972)  pp. 171–175</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  G.K. Pedersen,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310151.png" />-algebras and their automorphism groups" , Acad. Press (1979)</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  K. Saitô,  "On the embedding as a double commutator in a Type I <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310152.png" />-algebra II"  ''Tôhoku Math. J.'' , '''26'''  (1974)  pp. 333–339</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  K. Saitô,  "A structure theory in the regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310153.png" />-completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310154.png" />-algebras"  ''J. London Math. Soc.'' , '''22'''  (1980)  pp. 549–548</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  K. Saitô,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310155.png" />-algebras with monotone convergence properties and examples by Takenouchi and Dyer"  ''Tôhoku Math. J.'' , '''31'''  (1979)  pp. 31–40</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  D. Sullivan,  B. Weiss,  J.D.M. Wright,  "Generic dynamics and monotone complete <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310156.png" />-algebras"  ''Trans. Amer. Math. Soc.'' , '''295'''  (1986)  pp. 795–809</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top">  J.D.M. Wright,  "On semi-finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310157.png" />-algebras"  ''Math. Proc. Cambridge Philos. Soc.'' , '''79'''  (1975)  pp. 443–445</TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top">  J.D.M. Wright,  "On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310158.png" />-algebras of finite type"  ''J. London Math. Soc.'' , '''12'''  (1976)  pp. 431–439</TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top">  J.D.M. Wright,  "Regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310159.png" />-completions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310160.png" />-algebras"  ''J. London Math. Soc.'' , '''12'''  (1976)  pp. 299–309</TD></TR><TR><TD valign="top">[a23]</TD> <TD valign="top">  J.D.M. Wright,  "Wild <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120310/a120310161.png" />-factors and Kaplansky–Rickart algebras"  ''J. London Math. Soc.'' , '''13'''  (1976)  pp. 83–89</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  S.K. Berberian,  "Baer $*$-rings" , Springer  (1972)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  B. Blackadar,  D. Handelman,  "Dimension functions and traces on $C ^ { * }$-algebras" ''J. Funct. Anal.'' , '''45'''  (1982)  pp. 297–340</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  E. Christensen,  G.K. Pedersen,  "Properly infinite $A W ^ { * }$-algebras are monotone sequentially complete"  ''Bull. London Math. Soc.'' , '''16'''  (1984)  pp. 407–410</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  J. Dixmier,  "Sur certains espace considérés par M.H. Stone"  ''Summa Brasil. Math.'' , '''2'''  (1951)  pp. 151–182</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  G.A. Elliott,  K. Saitô,  J.D.M. Wright,  "Embedding $A W ^ { * }$-algebras as double commutants in Type I algebras"  ''J. London Math. Soc.'' , '''28'''  (1983)  pp. 376–384</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  M. Hamana,  "Regular embeddings of $C ^ { * }$-algebras in monotone complete $C ^ { * }$-algebras" ''J. Math. Soc. Japan'' , '''33'''  (1981)  pp. 159–183</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  D. Handelman,  "Homomorphisms of $C ^ { * }$-algebras to finite $A W ^ { * }$-algebras"  ''Michigan Math. J.'' , '''28'''  (1981)  pp. 229–240</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  R.V. Kadison,  G.K. Pedersen,  "Equivalence in operator algebras" ''Math. Scand.'' , '''27'''  (1970)  pp. 205–222</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  I. Kaplansky,  "Projections in Banach algebras"  ''Ann. Math.'' , '''53'''  (1951)  pp. 235–249</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  I. Kaplansky,  "Algebras of Type I"  ''Ann. Math.'' , '''56'''  (1952)  pp. 460–472</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  I. Kaplansky,  "Modules over operator algebras"  ''Amer. J. Math.'' , '''75'''  (1953)  pp. 839–858</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  I. Kaplansky,  "Rings of operators" , Benjamin (1968)</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  M. Ozawa,  "Nonuniqueness of the cardinality attached to homogeneous $A W ^ { * }$-algebras"  ''Proc. Amer. Math. Soc.'' , '''93'''  (1985)  pp. 681–684</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  G.K. Pedersen,  "Operator algebras with weakly closed Abelian subalgebras''Bull. London Math. Soc.'' , '''4'''  (1972)  pp. 171–175</td></tr><tr><td valign="top">[a15]</td> <td valign="top">  G.K. Pedersen,  "$C ^ { * }$-algebras and their automorphism groups" , Acad. Press  (1979)</td></tr><tr><td valign="top">[a16]</td> <td valign="top">  K. Saitô,  "On the embedding as a double commutator in a Type I $A W ^ { * }$-algebra II"  ''Tôhoku Math. J.'' , '''26'''  (1974)  pp. 333–339</td></tr><tr><td valign="top">[a17]</td> <td valign="top">  K. Saitô,  "A structure theory in the regular $\sigma$-completion of $C ^ { * }$-algebras"  ''J. London Math. Soc.'' , '''22'''  (1980)  pp. 549–548</td></tr><tr><td valign="top">[a18]</td> <td valign="top">  K. Saitô,  "$A W ^ { * }$-algebras with monotone convergence properties and examples by Takenouchi and Dyer" ''Tôhoku Math. J.'' , '''31''' (1979) pp. 31–40</td></tr><tr><td valign="top">[a19]</td> <td valign="top">  D. Sullivan,  B. Weiss,  J.D.M. Wright,  "Generic dynamics and monotone complete $C ^ { * }$-algebras"  ''Trans. Amer. Math. Soc.'' , '''295'''  (1986)  pp. 795–809</td></tr><tr><td valign="top">[a20]</td> <td valign="top">  J.D.M. Wright,  "On semi-finite $A W ^ { * }$-algebras" ''Math. Proc. Cambridge Philos. Soc.'' , '''79'''  (1975)  pp. 443–445</td></tr><tr><td valign="top">[a21]</td> <td valign="top">  J.D.M. Wright,  "On $A W ^ { * }$-algebras of finite type"  ''J. London Math. Soc.'' , '''12'''  (1976)  pp. 431–439</td></tr><tr><td valign="top">[a22]</td> <td valign="top">  J.D.M. Wright,  "Regular $\sigma$-completions of $C ^ { * }$-algebras" ''J. London Math. Soc.'' , '''12'''  (1976)  pp. 299–309</td></tr><tr><td valign="top">[a23]</td> <td valign="top">  J.D.M. Wright,  "Wild $A W ^ { * }$-factors and Kaplansky–Rickart algebras"  ''J. London Math. Soc.'' , '''13'''  (1976)  pp. 83–89</td></tr></table>

Latest revision as of 16:46, 1 July 2020

abstract von Neumann algebra

An algebra from a strictly larger class of $C ^ { * }$-algebras than the class of von Neumann algebras (cf. also von Neumann algebra). Such algebras were introduced by I. Kaplansky [a9], [a10], [a11], [a12], originally as a means of abstracting the algebraic properties of von Neumann algebras from their topological properties. Since von Neumann algebras are also known as $W ^ { * }$-algebras, such algebras were termed abstract $W ^ { * }$-algebras, or $A W ^ { * }$-algebras. Indeed the "classical" approach to $A W ^ { * }$-algebras was devoted to showing how closely their behaviour corresponded to that of von Neumann algebras. See [a1], and its extensive references, for a scholarly exposition of this classical material. However, in recent years, much effort has been devoted to investigating $A W ^ { * }$-algebras whose properties can be markedly different from their von Neumann cousins.

Let $M$ be a $C ^ { * }$-algebra with a unit element. Let $M _ { \operatorname{sa} }$ be the set of self-adjoint elements of $M$. Then $M _ { \operatorname{sa} }$ has a natural partial ordering which organizes $M _ { \operatorname{sa} }$ as a partially ordered real vector space with order-unit $1$ (cf. also Semi-ordered space). The positive cone of $M _ { \operatorname{sa} }$ for this partial ordering is the set of all elements of the form $zz ^ { * }$. When each upper bounded, upward-directed subset of $M _ { \operatorname{sa} }$ has a least upper bound, then $M$ is said to be monotone complete. All von Neumann algebras are monotone complete but the converse is false. To see this, it suffices to give examples of commutative $C ^ { * }$-algebras which are monotone complete but which are not von Neumann algebras.

An $A W ^ { * }$-algebra is a $C ^ { * }$-algebra $A$, with a unit, such that each maximal commutative $*$-subalgebra of $A$ is monotone complete. Clearly each monotone complete $C ^ { * }$-algebra is an $A W ^ { * }$-algebra and every commutative $A W ^ { * }$-algebra is monotone complete. It is natural to ask if every $A W ^ { * }$-algebra is monotone complete. Despite the important advances of [a3] this question is not yet (1999) settled.

Each commutative unital $C ^ { * }$-algebra $A$ is $*$-isomorphic to $C ( E )$, the $*$-algebra of all complex-valued continuous functions on a compact Hausdorff space $E$. Then the commutative algebra $A$ is an $A W ^ { * }$-algebra precisely when $E$ is extremally disconnected, that is, the closure of each open subset of $E$ is open (cf. also Extremally-disconnected space). It follows from the Stone representation theorem for Boolean algebras (cf. also Boolean algebra) that the projections in a commutative $A W ^ { * }$-algebra form a complete Boolean algebra and, conversely, all complete Boolean algebras arise in this way.

Let $K$ be a topological space which is homeomorphic to a complete separable metric space with no isolated points; let $B ( K )$ be the $*$-algebra of all bounded, Borel measurable, complex-valued functions on $K$. Let $M ( K )$ be the ideal of $B ( K )$ consisting of all functions $f$ for which $\{ x \in X : f ( x ) \neq 0 \}$ is meagre, that is, of first Baire category (cf. also Baire classes). Then $B ( K ) / M ( K )$ is a commutative monotone complete $C ^ { * }$-algebra which is isomorphic to $C ( S )$, where $S$ is a compact extremally disconnected space. The algebra $C ( S )$, which is independent of the choice of $K$, is known as the Dixmier algebra. It can be shown that $C ( S )$ has no states which are normal. It follows from this that $C ( S )$ is not a von Neumann algebra.

The classification of von Neumann algebras into Type I, Type-II, and Type-III (cf. also von Neumann algebra) can be extended to give a similar classification for $A W ^ { * }$-algebras. Let $B$ be a Type-I $A W ^ { * }$-algebra and let $A$ be an $A W ^ { * }$-algebra embedded as a subalgebra of $B$. If $A$ contains the centre of $B$ and if the lattice of projections of $A$ is a complete sublattice of the lattice of projections of $B$, then K. Saitô [a16] proved that $A$ equals its bi-commutant in $B$. This result extends earlier results by J. Feldman and by H. Widom and builds on the elegant characterization by G.K. Pedersen of von Neumann algebras [a14], [a15]. See also [a5]. By contrast, M. Ozawa [a13] showed that Type-I $A W ^ { * }$-algebras can exhibit pathological properties.

An $A W ^ { * }$-algebra $A$ is said to be an $A W ^ { * }$-factor if $A$ has trivial centre, that is,

\begin{equation*} \{ z \in A : z a = a z \;\text { for each } a \in A \} \end{equation*}

is one-dimensional. An early result of I. Kaplansky showed that each $A W ^ { * }$-factor of Type I was, in fact, a von Neumann algebra. This made it reasonable for him to ask if the same were true for $A W ^ { * }$-factors of Type II and Type III. For Type II there are partial results, described below, which make it plausible to conjecture that all Type-II $A W ^ { * }$-factors are von Neumann algebras. If this could be established then this would have important implications for separable $C ^ { * }$-algebras [a2], [a7]. For Type-III $A W ^ { * }$-factors the situation is completely different. Examples of such factors which are not von Neumann algebras are described below.

Let $A$ be an $A W ^ { * }$-factor of Type $\operatorname{II} _ { 1 }$. Then it was shown in [a21] that if $A$ possesses a faithful state, then $A$ possesses a faithful normal state and hence is a von Neumann factor of Type $\operatorname{II} _ { 1 }$. It follows from this that when $B$ is an $A W ^ { * }$-factor of Type II which possesses a faithful state, then $B$ is a von Neumann algebra [a5], [a20]. By contrast, there exist monotone complete $A W ^ { * }$-factors of Type III which possess faithful states but which are not von Neumann algebras.

Let $G$ be a countable group of homeomorphisms of a topological space $K$, where $K$ is homeomorphic to a complete separable metric space with no isolated points. Let the action of $G$ be free and let there exist a dense $G$-orbit. The action of $G$ on $K$ induces a free, generically ergodic action (of $G$ on $B ( K ) / M ( K ) = C ( S )$, the Dixmier algebra). Then there exists a corresponding cross product algebra $M ( C ( S ) , \alpha , G )$ which is a monotone complete $A W ^ { * }$-factor of Type III. Since this algebra contains a maximal commutative $*$-subalgebra isomorphic to $C ( S )$, which is not a von Neumann algebra, $M ( C ( S ) , \alpha , G )$ is not a von Neumann algebra. The first examples of Type-III factors which were not von Neumann algebras were constructed, independently, by O. Takenouchi and J. Dyer. Their respective examples were of the form $M ( C ( S ) , \alpha _ { 1 } , G _ { 1 } )$ and $M ( C ( S ) , \alpha _ { 2 } , G _ { 2 } )$ for (different) Abelian groups $G_1$ and $G_2$, see [a18]. As a corollary of the Sullivan–Weiss–Wright theorem [a19], the Takenouchi and Dyer factors are isomorphic. Much more is true. The algebra $M ( C ( S ) , \alpha , G )$ is independent of the choice of $G$ and $\alpha$ provided the action of $G$ is free and generically ergodic. For example, if one takes $G_1$ to be the additive group of integers and $G_2$ to be the free group on two generators the corresponding $A W ^ { * }$-factors are isomorphic. This is surprisingly different from the situation for von Neumann algebras. For a particularly lucid account of monotone cross-products see [a18].

Another approach to constructing monotone complete Type-III $A W ^ { * }$-factors which are not von Neumann algebras goes as follows. Let $A$ be a unital $C ^ { * }$-algebra, let $A ^ { \infty }$ be the Pedersen–Borel-$*$ envelope of $A$ on the universal representation space of $A$ [a15]. Then there is a "meagre" ideal $M$ in $A ^ { \infty }$ such that the quotient $A ^ { \infty } / M$ is a monotone $\sigma$-complete $C ^ { * }$-algebra $\hat{A}$ in which $A$ is embedded as an order-dense subalgebra. When $A$ is separable, simple and infinite dimensional, then $\hat{A}$ is a monotone complete $A W ^ { * }$-factor of Type III which is never a von Neumann algebra [a22], [a23]. This type of completion has been extensively generalized by M. Hamana [a6].

Although much progress has been made in understanding $A W ^ { * }$-factors, many unsolved problems remain.

References

[a1] S.K. Berberian, "Baer $*$-rings" , Springer (1972)
[a2] B. Blackadar, D. Handelman, "Dimension functions and traces on $C ^ { * }$-algebras" J. Funct. Anal. , 45 (1982) pp. 297–340
[a3] E. Christensen, G.K. Pedersen, "Properly infinite $A W ^ { * }$-algebras are monotone sequentially complete" Bull. London Math. Soc. , 16 (1984) pp. 407–410
[a4] J. Dixmier, "Sur certains espace considérés par M.H. Stone" Summa Brasil. Math. , 2 (1951) pp. 151–182
[a5] G.A. Elliott, K. Saitô, J.D.M. Wright, "Embedding $A W ^ { * }$-algebras as double commutants in Type I algebras" J. London Math. Soc. , 28 (1983) pp. 376–384
[a6] M. Hamana, "Regular embeddings of $C ^ { * }$-algebras in monotone complete $C ^ { * }$-algebras" J. Math. Soc. Japan , 33 (1981) pp. 159–183
[a7] D. Handelman, "Homomorphisms of $C ^ { * }$-algebras to finite $A W ^ { * }$-algebras" Michigan Math. J. , 28 (1981) pp. 229–240
[a8] R.V. Kadison, G.K. Pedersen, "Equivalence in operator algebras" Math. Scand. , 27 (1970) pp. 205–222
[a9] I. Kaplansky, "Projections in Banach algebras" Ann. Math. , 53 (1951) pp. 235–249
[a10] I. Kaplansky, "Algebras of Type I" Ann. Math. , 56 (1952) pp. 460–472
[a11] I. Kaplansky, "Modules over operator algebras" Amer. J. Math. , 75 (1953) pp. 839–858
[a12] I. Kaplansky, "Rings of operators" , Benjamin (1968)
[a13] M. Ozawa, "Nonuniqueness of the cardinality attached to homogeneous $A W ^ { * }$-algebras" Proc. Amer. Math. Soc. , 93 (1985) pp. 681–684
[a14] G.K. Pedersen, "Operator algebras with weakly closed Abelian subalgebras" Bull. London Math. Soc. , 4 (1972) pp. 171–175
[a15] G.K. Pedersen, "$C ^ { * }$-algebras and their automorphism groups" , Acad. Press (1979)
[a16] K. Saitô, "On the embedding as a double commutator in a Type I $A W ^ { * }$-algebra II" Tôhoku Math. J. , 26 (1974) pp. 333–339
[a17] K. Saitô, "A structure theory in the regular $\sigma$-completion of $C ^ { * }$-algebras" J. London Math. Soc. , 22 (1980) pp. 549–548
[a18] K. Saitô, "$A W ^ { * }$-algebras with monotone convergence properties and examples by Takenouchi and Dyer" Tôhoku Math. J. , 31 (1979) pp. 31–40
[a19] D. Sullivan, B. Weiss, J.D.M. Wright, "Generic dynamics and monotone complete $C ^ { * }$-algebras" Trans. Amer. Math. Soc. , 295 (1986) pp. 795–809
[a20] J.D.M. Wright, "On semi-finite $A W ^ { * }$-algebras" Math. Proc. Cambridge Philos. Soc. , 79 (1975) pp. 443–445
[a21] J.D.M. Wright, "On $A W ^ { * }$-algebras of finite type" J. London Math. Soc. , 12 (1976) pp. 431–439
[a22] J.D.M. Wright, "Regular $\sigma$-completions of $C ^ { * }$-algebras" J. London Math. Soc. , 12 (1976) pp. 299–309
[a23] J.D.M. Wright, "Wild $A W ^ { * }$-factors and Kaplansky–Rickart algebras" J. London Math. Soc. , 13 (1976) pp. 83–89
How to Cite This Entry:
AW*-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=AW*-algebra&oldid=49990
This article was adapted from an original article by J.D.M. Wright (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article