Definitions

central-projection

Central carrier

In the context of von Neumann algebras, the central carrier of a projection E is the smallest central projection, in the von Neumann algebra, that dominates E. It is also called the central supporting projection or central cover.

Definition

Let L(H) denote the bounded operators on a Hilbert space H, ML(H) be a von Neumann algebra, and M` the commutant of M. The center of M is Z(M) = M`M = {TM | TM = MT for all MM}. The central carrier C(E) of a projection E in M is defined as follows:

C(E) = ∧ {FZ(M) | F is a projection and FE}.

The symbol ∧ denotes the lattice operation on the projections in Z(M): F1F2 is the projection onto the closed subspace generated by Ran(F1) ∩ Ran(F2).

The abelian algebra Z(M), being the intersection of two von Neumann algebras, is also a von Neumann algebra. Therefore C(E) lies in Z(M).

If one think of M as a direct sum (or more accurately, a direct integral) of its factors, then the central projections are the direct sums of identity operators in the factors. If E is confined to a single factor, then C(E) is the identity operator in that factor. Informally, one would expect C(E) to be the direct sum of identity operators I where I is in a factor and I · E ≠ 0.

An explicit description

The projection C(E) can be described more explicitly. It can be shown that the Ran C(E) is the closed subspaces generated by MRan(E).

If N is a von Neumann algebra, and E a projection that does not necessarily belong to N and has range H`. The smallest central projection in N that dominates E is precisely the projection onto the closed subspace [N`H`] generated by N`H`. In symbols, if

F' = ∧ {FN | F is a projection and FE}

then Ran(F`) = [N`H`]. That [N`H`] ⊂ Ran(F`) follows from the definition of commutant. On the other hand, [N`H`] is invariant under every unitary U in N`. Therefore the projection onto [N`H`] lies in N. Minimality of F` then yields Ran(F`) ⊂ [N`H`].

Now if E is a projection in M, applying the above to the von Neumann algebra Z(M) gives

Ran C(E) = [Z(M)` Ran(E) ] = [(M`M)` Ran(E) ] = [MRan(E)].

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.
⇔ [M Ran(F)] ⊂ 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*UF.

In particular, when M is a factor, then there exists a partial isometry UM such that UU*E and U*UF. 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, FM 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 EiE, FiF, 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, FM are projections, then there exists a central projection PZ(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.

References

  • B. Blackadar, Operator Algebras, Springer, 2006.
  • S. Sakai, C*-Algebras and W*-Algebras, Springer, 1998.
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