Commutation Theorem

The commutation theorem of Murray and von Neumann states that

One of the easiest ways to see this is to introduce K, the closure of the real subspace M_sa Ω, where M_sa denotes the self-adjoint elements in M. It follows that

an orthogonal direct sum for the real part of inner product. This is just the real orthogonal decomposition for the ±1 eigenspaces of J. On the other hand for a in M_sa and b in M'_sa, the inner product (abΩ, Ω) is real, because ab is self-adjoint. Hence K is unaltered if M is replaced by M '.

In particular Ω is a trace vector for M' and J is unaltered if M is replaced by M '. So the opposite inclusion

follows by reversing the roles of M and M'.