Von Neumann algebras as complemented subspaces of $B({\mathscr{H}})$

Abstract

Let [Formula: see text] be a von Neumann algebra of type II 1 which is also a complemented subspace of [Formula: see text]. We establish an algebraic criterion, which ensures that [Formula: see text] is an injective von Neumann algebra. As a corollary we show that if [Formula: see text] is a complemented factor of type II 1 on a Hilbert space [Formula: see text], then [Formula: see text] is injective if its fundamental group is nontrivial.