Abelian projection - Imported Knowlpedia Content

Definition

Let $M$ be a von Neumann algebra. A projection $p\in M$ is an abelian projection if the corner $pMp$ , regarded as a von Neumann algebra with unit $p$ , is abelian. Equivalently,

xy=yx\qquad\text{for all }x,y\in pMp,

or the center of $pMp$ is all of $pMp$ . The definition concerns the compressed algebra, not merely the one-dimensional algebra generated by $p$ . In particular, $p$ need not be central in $M$ , and an abelian projection can have nontrivial central support.

Relation to minimal projections

In a factor, a nonzero projection is abelian exactly when it is minimal: indeed, the center of $pMp$ is scalar, so an abelian corner must equal $\mathbb Cp$ . Outside factors, an abelian projection need not be minimal; the commutative corner may contain many projections arising from its own center.

Type I criterion

A von Neumann algebra is type I precisely when every nonzero central projection dominates a nonzero abelian projection. Equivalently, its central summands are generated by abelian projections. For a factor, this reduces to the existence of one nonzero abelian projection, leading to the form $B(H)$ .

Examples and nonexamples

Every projection in an abelian von Neumann algebra is abelian. In $B(H)$ , the abelian projections are exactly the projections of rank at most one. By contrast, a type II factor has no nonzero abelian projections, even though it has many nonzero finite projections. Thus “abelian” and “finite” describe different features of a corner.

References

Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, Volume II, American Mathematical Society, 1997. Publisher record. Relevant: §6.4 on abelian projections and type I von Neumann algebras.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 2002. Publisher record. Relevant: Chapter V on the type classification.