On a canonical lattice structure on the effect algebra of a von Neumann algebra
نویسنده
چکیده
Let R be a von Neumann algebra acting on a Hilbert space H and let Rsa be the set of hermitean (i.e. selfadjoint) elements of R. It is well known that Rsa is a lattice with respect to the usual partial order ≤ if and only if R is abelian. We define and study a new partial order on Rsa, the spectral order ≤s, which extends ≤ on projections, is coarser than the usual one, but agrees with it on abelian subalgebras, and turns Rsa into a boundedly complete lattice. The effect algebra E(R) := {A ∈ Rsa|0 ≤ A ≤ I} is then a complete lattice and we show that the mapping A 7→ R(A), where R(A) denotes the range projection of A, is a homomorphism from the lattice E(R) onto the lattice P(R) of projections if and only if R is a finite von Neumann algebra. ∗e-mail: [email protected]
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