Applications of the complex interpolation method to a von Neumann algebra: non-commutative \(L^ p\)-spaces (Q1083667)

The paper includes a new construction of non-commutative \(L^ p\)-spaces (for \(1<p<\infty)\) associated with a von Neumann algebra which is not necessarily semi-finite. The construction is essentially different from that in the papers by Connes, Haagerup. The \(L^ p\)-spaces are defined as interpolation spaces between the von Neumann algebra and its predual. The uniform convexity of \(L^ p\)-spaces and the duality between \(L^ p\) and \(L^ q\) are proved by making use of the complex interpolation method and Tomita-Takesaki modular theory. Then, these \(L^ p\)-spaces are compared with Haagerup \(L^ p\)-spaces.