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

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