Is there any description of unital idempotent ($F^2(x)=F(x)$) morphisms of a von Neumann algebra into itself? Or, equivalently, of weakly closed subalgebras which are retracts as von Neumann algebras?



Idempotent homomorphisms of von Neumann algebras
Yes. The kernel of F is an ultraweakly closed \*-ideal of M generated by some central projection z.
M splits as a direct sum of zM and (1-z)M.
As a 2x2 matrix F has only two nonzero entries, one that corresponds to an idempotent automorphism (hence the identity map) of (1-z)M and another one to an arbitrary morphism from (1-z)M to zM.
Thus idempotent morphisms are classified by central projections and morphisms from (1-z)M to zM.


