Abstract
Let a von Neumann algebra \(\mathcal{M}\) of operators act on a Hilbert space \(\mathcal{H}\) , \(I\) be the unit of \(\mathcal{M}\) , \(\tau\) be a faithful semifinite normal trace on \(\mathcal{M}\) . Let \(S(\mathcal{M},\tau)\) be the \({}^{*}\) -algebra of all \(\tau\) -measurable operators and \(L_{1}(\mathcal{M},\tau)\) be the Banach space of all \(\tau\) -integrable operators, \(P,Q\in S(\mathcal{M},\tau)\) be idempotents. If \(P-Q\in L_{1}(\mathcal{M},\tau)\) then \(\tau(P-Q)\in\mathbb{R}\) . In particular, if \(A=A^{3}\in L_{1}(\mathcal{M},\tau)\) , then \(\tau(A)\in\mathbb{R}\) . If \(P-Q\in L_{1}(\mathcal{M},\tau)\) and \(PQ\in\mathcal{M}\) , then for all \(n\in\mathbb{N}\) we have \((P-Q)^{2n+1}\in L_{1}(\mathcal{M},\tau)\) and \(\tau((P-Q)^{2n+1})=\tau(P-Q)\in\mathbb{R}\) . If \(A\in L_{2}(\mathcal{M},\tau)\) and \(U\in\mathcal{M}\) is an isometry, then \(||UA-A||_{2}^{2}\leq 2||(I-U)AA^{*}||_{1}\) .
