Nonassociative Lp-spaces and embeddings in noncommutative Lp-spaces

Abstract

We define a notion of nonassociative Lp-space associated to a JBW⁎-algebra (Jordan von Neumann algebra) equipped with a normal faithful state φ. In the particular case of JW⁎-algebras underlying von Neumann algebras, we connect these spaces to a complex interpolation theorem of Ricard and Xu on noncommutative Lp-spaces. We also make the link with the nonassociative Lp-spaces of Iochum associated to JBW-algebras and the investigation of contractively complemented subspaces of noncommutative Lp-spaces. More precisely, we show that our nonassociative Lp-spaces contain isometrically the Lp-spaces of Iochum and that all tracial nonassociative Lp-spaces from JW⁎-factors arise as positively contractively complemented subspaces of noncommutative Lp-spaces.