On idempotent τ-measurable operators affiliated to a von Neumann algebra

Abstract

Let τ be a faithful normal semifinite trace on a von Neumann algebra M, let p, 0 < p < ∞, be a number, and let L _p (M, τ) be the space of operators whose pth power is integrable (with respect to τ). Let P and Q be τ-measurable idempotents, and let A ≡ P − Q. In this case, 1) if A ≥ 0, then A is a projection and QA = AQ = 0; 2) if P is quasinormal, then P is a projection; 3) if Q ∈ M and A ∈ L _p (M, τ), then A ² ∈ L _p (M, τ). Let n be a positive integer, n > 2, and A = A ⁿ ∈ M. In this case, 1) if A ≠ 0, then the values of the nonincreasing rearrangement μ _t (A) belong to the set {0} ∪ [‖A ⁿ⁻²‖⁻¹, ‖A‖] for all t > 0; 2) either μ _t (A) ≥ 1 for all t > 0 or there is a t ₀ > 0 such that μ _t (A) = 0 for all t > t ₀. For every τ-measurable idempotent Q, there is aunique rank projection P ∈ M with QP = P, PQ = Q, and PM = QM. There is a unique decomposition Q = P + Z, where Z ² = 0, ZP = 0, and PZ = Z. Here, if Q ∈ L _p (M, τ), then P is integrable, and τ(Q) = τ(P) for p = 1. If A ∈ L ₁(M, τ) and if A = A ³ and A − A ² ∈ M, then τ(A) ∈ R.