AW*-algebra

In mathematics, an AW*-algebra is an algebraic generalization of a W*-algebra. They were introduced by Irving Kaplansky in 1951.^[1] As operator algebras, von Neumann algebras, among all C*-algebras, are typically handled using one of two means: they are the dual space of some Banach space, and they are determined to a large extent by their projections. The idea behind AW*-algebras is to forgo the former, topological, condition, and use only the latter, algebraic, condition.

Definition

Recall that a projection of a C*-algebra is a self-adjoint idempotent element.

A C*-algebra A is an AW*-algebra if for every subset S of A, the right annihilator

$\mathrm{Ann}_R(S)=\{a\in A\mid \forall s\in S, as=0 \}\,$

is generated as a left ideal by some projection p of A, and similarly the left annihilator is generated as a right ideal by some projection q:

$\forall S \subseteq A\, \exists p,q \in \mathrm{Proj}(A) \colon \mathrm{Ann}_R(S)=Ap, \quad \mathrm{Ann}_L(S)=qA$.

Hence an AW*-algebra is a C*-algebras that is at the same time a Baer *-ring.

Structure theory

Many results concerning von Neumann algebras carry over to AW*-algebras. For example, AW*-algebras can be classified according to the behavior of their projections, and decompose into types.^[2] For another example, normal matrices with entries in an AW*-algebra can always be diagonalized.^[3] AW*-algebras also always have polar decomposition.^[4]

However, there are also ways in which AW*-algebras behave differently from von Neumann algebras.^[5] For example, AW*-algebras of type I can exhibit pathological properties,^[6] even though Kaplansky already showed that such algebras with trivial center are automatically von Neumann algebras.

The commutative case

A commutative C*-algebra is an AW*-algebra if and only if its spectrum is a Stonean space. Via Stone duality, commutative AW*-algebras therefore correspond to complete Boolean algebras. The projections of a commutative AW*-algebra form a complete Boolean algebra, and conversely, any complete Boolean algebra is isomorphic to the projections of some commutative AW*-algebra.

References

1. Kaplansky, Irving (1951). "Projections in Banach algebras". Annals of Mathematics 53 (2): 235–249. doi:10.2307/1969540.
2. Berberian, Sterling (1972). Baer *-rings. Springer.
3. Heunen, Chris; Reyes, Manuel L. (2013). "Diagonalizing matrices over AW*-algebras". Journal of Functional Analysis 264 (8): 1873–1898. doi:10.1016/j.jfa.2013.01.022.
4. Ara, Pere (1989). "Left and right projections are equivalent in Rickart C*-algebras". Journal of Algebra 120 (2): 433–448. doi:10.1016/0021-8693(89)90209-3.
5. Wright, J. D. Maitland. "AW*-algebra". Springer.
6. Ozawa, Masanao (1984). "Nonuniqueness of the cardinality attached to homogeneous AW*-algebras". Proceedings of the American Mathematical Society 93: 681–684. doi:10.2307/2045544.