Complemented subspaces of spaces of multilinear forms and tensor products. II. Noncommutative \(L_p\) spaces
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 \(S_{\hat r}^n\).
scientific article; zbMATH DE number 6387140
scientific article