Central Carrier - Definition

Definition

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:

C(E) = ∧ {F ∈ Z(M) | F is a projection and F ≥ E}.

The symbol ∧ denotes the lattice operation on the projections in Z(M): F₁ ∧ F₂ 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.