On subspaces of non-commutative \(L_{p}\)-spaces.
The main purpose of the paper under review is to investigate structural aspects of subspaces of non-commutative (Haagerup) \(L^p\)-spaces associated with general von Neumann algebras. One of the main results is on uniform containment of finite-dimensional Schatten classes \(S_p^n\) which follows from a more general theorem that goes as follows: Let \(0<p<\infty\), \(p\neq 2\), \(\mathcal{A}\) be a von Neumann algebra and \(X\) be a subspace of \(L^p(\mathcal{A})\). Let \((F_n)\) be a sequence of finite-dimensional (quasi-)normed spaces. If \(X\) contains \(K\)-uniformly \(\ell_p^n(F_j)\) for \(n,j\geq 1\), then \(X\) contains a subspace isomorphic to \(\left( \oplus_j F_j\right)_p\). For \(1\leq p<\infty\), a complemented version of the above stated result also holds. This result has a forerunner in the case of \(\mathcal{A}=B(\ell_2)\) and \(F_n=S_p^n\) in [\textit{J. Arazy} and \textit{J. Lindenstrauss}, Compos. Math. 30, 81--111 (1975; Zbl 0302.47034)]. The main tool is the concrete description of ultraproducts of non-commutative \(L^p\)-spaces developed by the first author in [J. Oper. Theory 48, No. 1, 41--68 (2002; Zbl 1029.46102)]. More precisely, the authors carefully study elements of the ultraproduct \(L^p(\mathcal{A})_{\mathcal{U}}\) which are disjoint from \(L^p(\mathcal{A})\). Their techniques allow them to deduce several results that in turn provide the following version of a non-commutative analogue of the classical Kadec-Pełczyński dichotomy (which was only known previously for the semi-finite case): Let \(\mathcal{A}\) be a von Neumann algebra, \(2<p<\infty\), and \(X\) be a closed subspace of \(L^p(\mathcal{A})\). Then either \(X\) is isomorphic to a Hilbert space and complemented in \(L^p(\mathcal{A})\) or \(X\) contains a subspace isomorphic to \(\ell_p\) and complemented in \(L^p(\mathcal{A})\). The authors also include a section that generalizes the results described above to the case of operator spaces.
scientific article; zbMATH DE number 1992840
scientific article