AW*-algebra

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