Return to Question

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

Edited: I do not claim anymore that these two questions are equivalent.

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?

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

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

Edited: I do not claim anymore that these two questions are equivalent.

Hope my question is simple. :)

Is there any description of unital idempotent ($F^2(x)=F(x)$) homomorphisms of a von Neumann algebra into itself? All I can think of is the identity morphism.

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