Unitary operators in real von Neumann algebras

We prove that the open unit ball of any von Neumann algebra A is contained in the sequentially convex hull of U A , the set of unitary elements of A . Therefore the closed unit ball of A coincides with the closed convex hull of U A . In the complex case this statement is actually valid for any unital C ⁎ -algebra (it is the well-known Russo–Dye Theorem). However, for real scalars, the results we are presenting provide new information even for the algebra L ( H ) of bounded linear operators from an infinite-dimensional Hilbert space H into itself. Let us say in this sense that the possibility of rebuilding the unit ball of L ( H ) through the closed convex hull of the unitary elements appears in the literature as an open problem. We also obtain some results about the extremal structure of these spaces.