Finite projection - Imported Knowlpedia Content

Definition

Let $p$ be a projection in a von Neumann algebra $M$ . It is a finite projection if, whenever $q\leq p$ and $q$ is Murray–von Neumann equivalent to $p$ , one has $q=p$ . Equivalently, there is no partial isometry $v\in M$ satisfying

v^*v=p,\qquad vv^*<p.

A projection that is not finite is called infinite. The zero projection is finite. Finiteness is intrinsic to the ambient von Neumann algebra and its partial isometries; it is not the same as finite-dimensionality of the projection's range in a particular representation.

Corners and equivalence

Finiteness is invariant under Murray–von Neumann equivalence and passes to subprojections. The projection $p$ is finite exactly when the corner $pMp$ , whose identity is $p$ , is a finite von Neumann algebra. A von Neumann algebra is called finite when its identity projection is finite.

Examples

In $B(H)$ , a projection is finite exactly when its range is finite-dimensional. In a type $II_1$ factor, every projection is finite, including projections with infinite-dimensional Hilbert-space range. The identity of $B(H)$ for infinite-dimensional $H$ is infinite, as a unilateral shift implements equivalence with a proper subprojection.

Finite versus properly infinite

An infinite projection $p$ is properly infinite when it contains two orthogonal subprojections, each equivalent to $p$ . Not every infinite projection is properly infinite in an arbitrary von Neumann algebra, because finite and properly infinite behavior can coexist on different central summands. In a factor, the type classification sharply constrains these possibilities.

References

Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, Volume II, American Mathematical Society, 1997. Publisher record. Relevant: §6.3 on Murray–von Neumann equivalence and finite projections.
F. J. Murray and J. von Neumann, “On Rings of Operators,” Annals of Mathematics 37 (1936), 116–229. JSTOR record. Relevant: comparison of projections and the finite/infinite distinction.