Concerning the theory of τ-measurable operators affiliated to a semifinite von Neumann algebra

Let M be a von Neumann algebra of operators in a Hilbert space H, let τ be an exact normal semifinite trace on M, and let L1(M, τ) be the Banach space of τ-integrable operators. The following results are obtained. If X = X*, Y = Y* are τ-measurable operators and XY ∈ L1(M, τ), then YX ∈ L1(M, τ) and τ(XY) = τ(YX) ∈ R. In particular, if X, Y ∈ B(H)sa and XY ∈ G1, then YX ∈ G1 and tr(XY) = tr(YX) ∈ R. If X ∈ L1(M, τ), then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau \left( {X*} \right) = \tau \left( X \right)$$\end{document}. Let A be a τ-measurable operator. If the operator A is τ-compact and V ∈ M is a contraction, then it follows from V* AV = A that V A = AV. We have A = A2 if and only if A = |A*||A|. This representation is also new for bounded idempotents in H. If A = A2 ∈ L1(M, τ), then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau \left( A \right) = \tau \left( {\sqrt {\left| A \right|} \left| {A*} \right|\sqrt {\left| A \right|} } \right) \in {\mathbb{R}^ + }$$\end{document}. If A = A2 and A (or A*) is semihyponormal, then A is normal, thus A is a projection. If A = A3 and A is hyponormal or cohyponormal, then A is normal, and thus A = A* ∈ M is the difference of two mutually orthogonal projections (A + A2)/2 and (A2 − A)/2. If A,A2 ∈ L1(M, τ) and A = A3, then τ(A) ∈ R.