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 support or central cover.
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}. The central carrier C(E) of a projection E in M is defined as follows:
The symbol ∧ denotes the lattice operation on the projections in Z(M): F1 ∧ F2 is the projection onto the closed subspace 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 thinks of M as a direct sum (or more accurately, a direct integral) of its factors, then the central projections are the projections that are direct sums (direct integrals) of identity operators of (measurable sets of) 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.
The projection C(E) can be described more explicitly. It can be shown that Ran C(E) is the closed subspace 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 K = Ran(E). The smallest central projection in N that dominates E is precisely the projection onto the closed subspace [N' K] generated by N' K. In symbols, if
then Ran(F' ) = [N' K]. That [N' K] ⊂ Ran(F' ) follows from the definition of commutant. On the other hand, [N' K] is invariant under every unitary U in N' . Therefore the projection onto [N' K] lies in (N')' = N. Minimality of F' then yields Ran(F' ) ⊂ [N' K].
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 turn, the following is true:
Corollary Two projections E and F in a von Neumann algebra M contain two nonzero sub-projections that are Murray-von Neumann equivalent if 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.
Proposition (Comparability) If M is a factor, and E, F ∈ M are projections, then either E « F or F « E.
Proof:
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: