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 support or central cover.

Definition

Let L denote the bounded operators on a Hilbert space H, M ⊂ L be a von Neumann algebra, and M' the commutant of M. The center of M is Z = M' ∩ M =. The central carrier C of a projection E in M is defined as follows:
The symbol ∧ denotes the lattice operation on the projections in Z: F1 ∧ F2 is the projection onto the closed subspace Ran ∩ Ran.
The abelian algebra Z, being the intersection of two von Neumann algebras, is also a von Neumann algebra. Therefore, C lies in Z.
If one thinks of M as a direct sum of its factors, then the central projections are the projections that are direct sums of identity operators of the factors. If E is confined to a single factor, then C is the identity operator in that factor. Informally, one would expect C 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 can be described more explicitly. It can be shown that Ran C is the closed subspace generated by MRan.
If N is a von Neumann algebra, and E a projection that does not necessarily belong to N and has range K = Ran. The smallest central projection in N that dominates E is precisely the projection onto the closed subspace generated by N' K. In symbols, if
then Ran = . That ⊂ Ran follows from the definition of commutant. On the other hand, is invariant under every unitary U in N' . Therefore the projection onto lies in ' = N. Minimality of F' then yields Ran ⊂ .
Now if E is a projection in M, applying the above to the von Neumann algebra Z gives

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 and C are orthogonal, i.e. C'C = 0.
Proof:
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'C ≠ 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 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 If M is a von Neumann algebra, and E, F ∈ M are projections, then there exists a central projection P ∈ Z such that either EP « FP and F « E.
Proof: