Commutation Theorem - Commutation Theorem For Semifinite Traces

Commutation Theorem For Semifinite Traces

Let M be a von Neumann algebra and M₊ the set of positive operators in M. By definition, a semifinite trace (or sometimes just trace) on M is a functional τ from M₊ into such that

1. for a, b in M₊ and λ, μ ≥ 0 (semilinearity);
2. for a in M₊ and u a unitary operator in M (unitary invariance);
3. τ is completely additive on orthogonal families of projections in M (normality);
4. each projection in M is as orthogonal direct sum of projections with finite trace (semifiniteness).

If in addition τ is non-zero on every non-zero projection, then τ is called a faithful trace.

If τ is a faithul trace on M, let H = L2(M, τ) be the Hilbert space completion of the inner product space

with respect to the inner product

The von Neumann algebra M acts by left multiplication on H and can be identified with its image. Let

for a in M₀. The operator J is again called the modular conjugation operator and extends to a conjugate-linear isometry of H satisfying J2 = I. The commutation theorem of Murray and von Neumann

is again valid in this case. This result can be proved directly by a variety of methods, but follows immediately from the result for finite traces, by repeated use of the following elementary fact:

If M₁ M₂ are two von Neumann algebras such that p_n M₁ = p_n M₂ for a family of projections p_n in the commutant of M₁ increasing to I in the strong operator topology, then M₁ = M₂.