On a property of \(L_p\) spaces on semifinite von Neumann algebras
type: Item
instance of: scholarly article
title: On a property of \(L_p\) spaces on semifinite von Neumann algebras
zbMATH Open document ID: 0927.46046
DOI: 10.1007/BF02310299
published in: Mathematical Notes
publication date: 1999-12-06T00:00:00Z
review text: 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 \( (M,\tau) \).
Mathematics Subject Classification ID: 46L51
zbMATH DE Number: 1274229
zbMATH Keywords: von Neumann algebra
zbMATH Keywords: weight
zbMATH Keywords: trace
zbMATH Keywords: measurable operators
zbMATH Keywords: \(L_p\)-spaces on semifinite algebras
zbMATH Keywords: Segal-measurable operators
zbMATH Keywords: \(\tau\)-integrable operators
zbMATH Keywords: rearrangement invariant spaces
zbMATH Keywords: normed ideal spaces
author: Airat Bikchentaev
reviewed by: Andrey V. Bulinski
MaRDI profile type: Publication
cites work: Q3766320
cites work: Q3876492
cites work: Q5653193
cites work: Q5515745
cites work: Non-commutative LP-spaces
cites work: Noncommutative Kothe Duality
cites work: Q4282390
cites work: Q5749818
cites work: Q4317514
full work available at URL: https://doi.org/10.1007/bf02310299
OpenAlex ID: W1979990741
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description: scientific article; zbMATH DE number 1274229