In mathematics, an AW*-algebra is an algebraic generalization of a W*-algebra. They were introduced by Irving Kaplansky in 1951. 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.
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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
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:
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. For another example, normal matrices with entries in an AW*-algebra can always be diagonalized. AW*-algebras also always have polar decomposition.
However, there are also ways in which AW*-algebras behave differently from von Neumann algebras. For example, AW*-algebras of type I can exhibit pathological properties, 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.