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 UA, the set of unitary elements of A. Therefore the closed unit ball of A coincides with the closed convex hull of UA. 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.