most recent 30 from http://mathoverflow.net 2013-06-20T11:14:44Z http://mathoverflow.net/feeds/question/100368 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/100368/idempotent-homomorphisms-of-von-neumann-algebras Yulia Kuznetsova 2012-06-22T15:27:44Z 2012-06-23T00:59:13Z

<p>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?</p>

http://mathoverflow.net/questions/100368/idempotent-homomorphisms-of-von-neumann-algebras/100382#100382 Dmitri Pavlov 2012-06-22T18:35:01Z 2012-06-22T18:35:01Z

<p>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.</p>