Complemented subspaces of spaces of multilinear forms and tensor products. II. Noncommutative \(L_p\) spaces
type: Item
instance of: scholarly article
title: Complemented subspaces of spaces of multilinear forms and tensor products. II. Noncommutative \(L_p\) spaces
zbMATH Open document ID: 1318.46028
author: Félix Cabello Sánchez
published in: Journal of Mathematical Analysis and Applications
publication date: 2015-01-16T00:00:00Z
review text: Given a Banach algebra \(A\) and quasi-Banach spaces \(X_1,\dots, X_k,W\) such that \(X_1,\dots,X_{k-1}\) are right modules over \(A\) and \(X_2,\dots, X_k\) are left modules over \(A\), a \(k\)-linear mapping \(B: X_1\times\cdots\times X_k\to W\) is said to be balanced if \[ B(x_1,\dots, x_ia,x_{i+1},\dots x_k)=B(x_1,\dots, x_i,ax_{i+1},\dots x_k ) \] for \(x_i\) in \(X_i\), \(1\leq i\leq k\), and \(a\) in \(A\). The author shows that, if \(L_{p_1},\dots,L_{p_k}\) are noncommutative \(L_p\) spaces in the sense of Haagerup with \(\sum_{i=1}^kp_i^{-1}=r^{-1}\) and \(B\) is a bounded, balanced \(k\)-linear mapping from \(L_{p_1}\times\cdots\times L_{p_k}\) into \(W\), then there is a bounded linear mapping \(T: L_r\to W\) such that \(B(f_1,\dots,f_k)=T(f_1 \dots f_k)\) and \(\|B\|=\|T\|\). Using this, he shows that, under certain conditions, the existence of a projection from the space of all continuous bilinear forms on \(L_2\times L_2\) onto the set of all continuous balanced bilinear forms on \(L_2\times L_2\) over a von Neumann algebra \(M\) implies that \(M\) is amenable. Using \(S_p^n\) to denote the \(n\)-dimensional Hilbert space \(H_n\) with the Schatten \(p\) (quasi-)norm, the author shows that, if \(p^{-1}+q^{-1}= r^{-1}\) and \(\hat r=\max\{1,r\}\), then there there is an isometric embedding of \(S_{\hat r}^n\) into \(S_p^n{\widehat\otimes} S_q^n\). From this, he proves that for \(\phi\) in \(B(H_n)\), the nuclear norm of the multiplication mapping, \(S_p^n\to S_q^n\), \(h\mapsto h\phi\), is \(n\) times the norm of \(\phi\) in \(S_{\hat r}^n\).
reviewed by: Christopher Boyd
Mathematics Subject Classification ID: 46G25
Mathematics Subject Classification ID: 46L52
zbMATH DE Number: 6387140
zbMATH Keywords: balanced multilinear mappings
zbMATH Keywords: Schatten \(p\)-class
zbMATH Keywords: amenable von Neumann algebra
zbMATH Keywords: multilinear forms
zbMATH Keywords: noncommutative \(L_p\) space
zbMATH Keywords: von Neumann algebra
MaRDI profile type: Publication
full work available at URL: j.jmaa.2014.11.030
OpenAlex ID: W1982737076
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DOI: 10.1016/J.JMAA.2014.11.030
description: scientific article; zbMATH DE number 6387140