AW*-algebra

[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.

A C*-algebra A is an AW*-algebra if for every subset S of A, the left annihilator is generated as a left ideal by some projection p of A, and similarly the right 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.

[4] However, there are also ways in which AW*-algebras behave differently from von Neumann algebras.

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.