Central Carrier - Related Results

Related Results

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:

ETF = 0 for all T in M.

⇔ ⊂ Ker(E).

⇔ C(F) ≤ 1 - E, by the discussion in the preceding section, where 1 is the unit in M.

⇔ E ≤ 1 - C(F).

⇔ C(E) ≤ 1 - C(F), since 1 - C(F) is a central projection that dominates E.

This proves the claim.

In turn, the following is true:

Corollary Two projections E and F in a von Neumann algebra M contain two nonzero subprojections that are Murray-von Neumann equivalent if C(E)C(F) ≠ 0.

Proof:

C(E)C(F) ≠ 0.

⇒ ETF ≠ 0 for some T in M.

⇒ ETF has polar decomposition UH for some partial isometry U and positive operator H in M.

⇒ Ran(U) = Ran(ETF) ⊂ Ran(E). Also, Ker(U) = Ran(H)⊥ = Ran(ETF)⊥ = Ker(ET*F) ⊃ Ker(F); therefore Ker(U))⊥ ⊂ Ran(F).

⇒ The two equivalent projections UU* and U*U satisfy UU* ≤ E and U*U ≤ F.

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.

Proposition (Comparability) If M is a factor, and E, F ∈ M are projections, then either E « F or F « E.

Proof:

Let ~ denote the Murray-von Neumann equivalence relation. Consider the family S whose typical element is a set { (E_i, F_i) } where the orthogonal sets {E_i} and {F_i} satisfy E_i ≤ E, F_i ≤ F, and E_i ~ F_i. The family S is partially ordered by inclusion and the above corollary shows it is non-empty. Zorn's lemma ensures the existence of a maximal element { (E_j, F_j) }. Maximality ensures that either E = ∑ E_j or F = ∑ F_j. The countable additivity of ~ means E_j ~ ∑ F_j. Thus the proposition holds.

Without the assumption that M is a factor, we have:

Proposition (Generalized Comparability) If M is a von Neumann algebra, and E, F ∈ M are projections, then there exists a central projection P ∈ Z(M) such that either EP « FP and F(1 - P) « E(1 - P).

Proof:

Let S be the same as in the previous proposition and again consider a maximal element { (E_j, F_j) }. Let R and S denote the "remainders": R = E - ∑ E_j and S = F - ∑ F_j. By maximality and the corollary, RTS = 0 for all T in M. So C(R)C(S) = 0. In particular R · C(S) = 0 and S · C(S) = 0. So multiplication by C(S) removes the remainder R from E while leaving S in F. More precisely, E · C(S) = (∑ E_j + R) · C(S) = (∑ E_j) · C(S) ~ (∑ F_j) · C(S) ≤ (∑ F_j + S) · C(S) = F · C(S). This shows that C(S) is the central projection with the desired properties.