An Explicit Description
The projection C(E) can be described more explicitly. It can be shown that the Ran C(E) is the closed subspaces 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 H`. The smallest central projection in N that dominates E is precisely the projection onto the closed subspace generated by N`H`. In symbols, if
- F' = ∧ {F ∈ N | F is a projection and F ≥ E}
then Ran(F`) = . That ⊂ Ran(F`) 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(F`) ⊂ .
Now if E is a projection in M, applying the above to the von Neumann algebra Z(M) gives
- Ran C(E) = = = .
Read more about this topic: Central Carrier
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