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 { (Ei, Fi) } where the orthogonal sets {Ei} and {Fi} satisfy Ei ≤ E, Fi ≤ F, and Ei ~ Fi. 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 { (Ej, Fj) }. Maximality ensures that either E = ∑ Ej or F = ∑ Fj. The countable additivity of ~ means Ej ~ ∑ Fj. 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 { (Ej, Fj) }. Let R and S denote the "remainders": R = E - ∑ Ej and S = F - ∑ Fj. 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) = (∑ Ej + R) · C(S) = (∑ Ej) · C(S) ~ (∑ Fj) · C(S) ≤ (∑ Fj + S) · C(S) = F · C(S). This shows that C(S) is the central projection with the desired properties.
Read more about this topic: Central Carrier
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