View source for AW*-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.

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.

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.

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.

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]].

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.

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,

<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>

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.

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.

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]]].

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]]].

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.

====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. 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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. 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