Embeddings of non-commutative \(L^p\)-spaces into preduals of finite von Neumann algebras
This paper studies the embedding problem of noncommutative \(L_p\)-spaces. The first main result states that, for any \(\sigma\)-finite von Neumann algebra \(R\), there exists a finite von Neumann algebra \(M\) such that \(L_p(R)\) isomorphically embeds into \(L_q(M)\) for any \(0<q<p<2\). The space \(L_p(R)\) used here is that defined by Haagerup. The author's proof heavily relies upon Haagerup's construction of \(L_p(R)\) as a space of measurable operators on a semifinite super-algebra. The second main result of the paper is the noncommutative analogue for semifinite algebras of Rosenthal's celebrated embedding theorem on reflexive subspaces of \(L_1\). Namely, if \(M\) is a semifinite von Neumann algebra and \(X\) a reflexive subspace of \(L_1(M)\), then \(X\) isomorphically embeds into \(L_p(M)\) for some \(1<p<2\). The key ingredient of the proof is an embedding theorem of \textit{G.\,Pisier} [Math.\ Ann.\ 276, 105--136 (1986; Zbl 0619.47016)]. Note that the noncommutative Rosenthal theorem in its full generality (for any von Neumann algebra) was recently proved by \textit{M.\,Junge} and \textit{J.\,Parcet} [Duke Math.\ J.\ 141, No.\,1, 75--122 (2008; Zbl 1176.46059)].
scientific article; zbMATH DE number 5279150
scientific article