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.
Read more about this topic: Central Carrier
Famous quotes containing the word definition:
“Im beginning to think that the proper definition of Man is an animal that writes letters.”
—Lewis Carroll [Charles Lutwidge Dodgson] (18321898)
“... we all know the wags definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on life (based on wording in the First Edition, 1935)