On a property of \(L_p\) spaces on semifinite von Neumann algebras
Let \( \varphi \) be a weight on a von Neumann algebra \( M \) such that \( \| x_n-x\| \to 0 \), \( x_n\nearrow x \), \( x,x_n\in M \), \( n\in\mathbb N\Rightarrow\varphi(x_n)\to\varphi(x) \), \( n\to\infty \) (say, \( \varphi \) is lower-semicontinuous with respect to the norm). Then \( \varphi \) is a trace \( \iff\varphi(ghg)=\varphi(hgh) \) for all projections \( g,h\in M \). For such a trace \( \varphi \), if \( x,y\in M^+ \) and \( xy\in\mathop{\text{Lin}}\mathfrak M_\varphi^+ \) then \( x^{1/2}yx^{1/2} \), \( y^{1/2}xy^{1/2}\in \mathfrak M_\varphi^+=\{z\in M^+, \varphi(z)<\infty\} \). The main claim refers to \( M \) with a faithful normal semifinite trace \( \tau \). Let \( K \) be the class of Segal-measurable operators affiliated with \( M \) and \( L_p=L_p(M,\tau) \) the corresponding spaces of \( \tau \)-integrable operators. If \( xy\in L_1 \) for \( x,y\in K^+ \) then \( \tau(xy)=\tau(x^{1/2}yx^{1/2})=\tau(y^{1/2}xy^{1/2}) \). This result extending that of Theorem 3.4 in [\textit{P. G. Dodds, T. K.-Y. Dodds} and \textit{B. de Pagter}, Trans. Am. Math. Soc. 339, No.~2, 717-750 (1993; Zbl 0801.46074)] is new even for (\( {\mathcal B}({\mathcal H}),tr \)). Some corollaries for \( L_p \), \( p\in\mathbb N \), are given. Moreover, the construction in loc. cit. of Koethe duality for rearrangement invariant spaces can be extended to a broader class of normed ideal spaces \(E\subset K \) over \( (M,\tau) \).
scientific article; zbMATH DE number 1274229