AW*-algebra

abstract von Neumann algebra

An algebra from a strictly larger class of -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 -algebras, such algebras were termed abstract -algebras, or -algebras. Indeed the "classical" approach to -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 -algebras whose properties can be markedly different from their von Neumann cousins.

Let be a -algebra with a unit element. Let be the set of self-adjoint elements of . Then has a natural partial ordering which organizes as a partially ordered real vector space with order-unit (cf. also Semi-ordered space). The positive cone of for this partial ordering is the set of all elements of the form . When each upper bounded, upward-directed subset of has a least upper bound, then 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 -algebras which are monotone complete but which are not von Neumann algebras.

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

Each commutative unital -algebra is -isomorphic to , the -algebra of all complex-valued continuous functions on a compact Hausdorff space . Then the commutative algebra is an -algebra precisely when is extremally disconnected, that is, the closure of each open subset of 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 -algebra form a complete Boolean algebra and, conversely, all complete Boolean algebras arise in this way.

Let be a topological space which is homeomorphic to a complete separable metric space with no isolated points; let be the -algebra of all bounded, Borel measurable, complex-valued functions on . Let be the ideal of consisting of all functions for which is meagre, that is, of first Baire category (cf. also Baire classes). Then is a commutative monotone complete -algebra which is isomorphic to , where is a compact extremally disconnected space. The algebra , which is independent of the choice of , is known as the Dixmier algebra. It can be shown that has no states which are normal. It follows from this that 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 -algebras. Let be a Type-I -algebra and let be an -algebra embedded as a subalgebra of . If contains the centre of and if the lattice of projections of is a complete sublattice of the lattice of projections of , then K. Saitô [a16] proved that equals its bi-commutant in . 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 -algebras can exhibit pathological properties.

An -algebra is said to be an -factor if has trivial centre, that is,

is one-dimensional. An early result of I. Kaplansky showed that each -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 -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 -factors are von Neumann algebras. If this could be established then this would have important implications for separable -algebras [a2], [a7]. For Type-III -factors the situation is completely different. Examples of such factors which are not von Neumann algebras are described below.

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

Let be a countable group of homeomorphisms of a topological space , where is homeomorphic to a complete separable metric space with no isolated points. Let the action of be free and let there exist a dense -orbit. The action of on induces a free, generically ergodic action (of on , the Dixmier algebra). Then there exists a corresponding cross product algebra which is a monotone complete -factor of Type III. Since this algebra contains a maximal commutative -subalgebra isomorphic to , which is not a von Neumann algebra, 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 and for (different) Abelian groups and , 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 is independent of the choice of and provided the action of is free and generically ergodic. For example, if one takes to be the additive group of integers and to be the free group on two generators the corresponding -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 -factors which are not von Neumann algebras goes as follows. Let be a unital -algebra, let be the Pedersen–Borel- envelope of on the universal representation space of [a15]. Then there is a "meagre" ideal in such that the quotient is a monotone -complete -algebra in which is embedded as an order-dense subalgebra. When is separable, simple and infinite dimensional, then is a monotone complete -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 -factors, many unsolved problems remain.

References

[a1]	S.K. Berberian, "Baer -rings" , Springer (1972)
[a2]	B. Blackadar, D. Handelman, "Dimension functions and traces on -algebras" J. Funct. Anal. , 45 (1982) pp. 297–340
[a3]	E. Christensen, G.K. Pedersen, "Properly infinite -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 -algebras as double commutants in Type I algebras" J. London Math. Soc. , 28 (1983) pp. 376–384
[a6]	M. Hamana, "Regular embeddings of -algebras in monotone complete -algebras" J. Math. Soc. Japan , 33 (1981) pp. 159–183
[a7]	D. Handelman, "Homomorphisms of -algebras to finite -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 -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, "-algebras and their automorphism groups" , Acad. Press (1979)
[a16]	K. Saitô, "On the embedding as a double commutator in a Type I -algebra II" Tôhoku Math. J. , 26 (1974) pp. 333–339
[a17]	K. Saitô, "A structure theory in the regular -completion of -algebras" J. London Math. Soc. , 22 (1980) pp. 549–548
[a18]	K. Saitô, "-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 -algebras" Trans. Amer. Math. Soc. , 295 (1986) pp. 795–809
[a20]	J.D.M. Wright, "On semi-finite -algebras" Math. Proc. Cambridge Philos. Soc. , 79 (1975) pp. 443–445
[a21]	J.D.M. Wright, "On -algebras of finite type" J. London Math. Soc. , 12 (1976) pp. 431–439
[a22]	J.D.M. Wright, "Regular -completions of -algebras" J. London Math. Soc. , 12 (1976) pp. 299–309
[a23]	J.D.M. Wright, "Wild -factors and Kaplansky–Rickart algebras" J. London Math. Soc. , 13 (1976) pp. 83–89