Nonassociative ${\mathrm{L}}^p$-spaces and embeddings in noncommutative ${\mathrm{L}}^p$-spaces

Abstract

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

26 pages, minor improvements and corrections