Central carrier
Let L(H) denote the bounded operators on a Hilbert space H, M ⊂ L(H) be a von Neumann algebra, and M' the commutant of M. The center of M is Z(M) = M' ∩ M = {T ∈ M | TM = MT for all M ∈ M}.If N is a von Neumann algebra, and E a projection that does not necessarily belong to N and has range K = Ran(E).Now if E is a projection in M, applying the above to the von Neumann algebra Z(M) gives One can deduce some simple consequences from the above description.Suppose E and F are projections in a von Neumann algebra M. Proposition ETF = 0 for all T in M if and only if C(E) and C(F) are orthogonal, i.e. C(E)C(F) = 0.Proof: In particular, when M is a factor, then there exists a partial isometry U ∈ M such that UU* ≤ E and U*U ≤ F. Using this fact and a maximality argument, it can be deduced that the Murray-von Neumann partial order « on the family of projections in M becomes a total order if M is a factor.