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:
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)