Von Neumann alternating projection theorem

The problem is to find an "expression" for the projection described. Although most mathematicians would read the statement of such a problem with sympathetic understanding, it must be admitted that rigorously speaking it does not really mean anything. The most obvious way to make it precise is to describe certain classes of operators by the requirement that they be closed under some familiar algebraic and topological operations, and then try to prove that whenever $E$ and $F$ belong to such a class, then so does $E \wedge F$. The most famous and useful classes pertinent here are the von Neumann algebras (called "rings of operators" by von Neumann). A von Neumann algebra is an algebra of operators (i.e., a collection closed under addition and multiplication, and closed under multiplication by arbitrary scalars), self-adjoint (i.e., closed under adjunction), containing $1$, and strongly closed (i.e., closed with respect to the strong operator topology). For von Neumann algebras, then, the problem is this: prove that if a von Neumann algebra contains two projections $E$ and $F$, then it contains $E \wedge F$.

Reference: [150, vol.2, p.55].