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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 forego the former, topological, condition, and use only the latter, algebraic, condition.

Definition

Recall that a projection of a C*-algebra $A$ is an element $p\in A$ satisfying $p^{*}=p=p^{2}$.

A C*-algebra $A$ is an AW*-algebra when for every subset $S\subseteq A$, 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. 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

By Gelfand duality, any commutative C*-algebra $A$ is isomorphic to the algebra of continuous functions $X\to \mathbb {C}$ for some compact Hausdorff space $X$. If $A$ is an AW*-algebra, then $X$ is in fact 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

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