Spectral versus interpolation norms in tracial nonassociative $\mathrm{L}^p$-spaces

Abstract

We investigate the metric structure of nonassociative $\mathrm{L}^p$-spaces associated with tracial $\mathrm{JW}^*$-algebras. While noncommutative $\mathrm{L}^p$-spaces arising from von Neumann algebras enjoy a unique natural norm, the situation in the Jordan setting is more subtle. We compare two canonical definitions: the interpolation norm, arising from the complex method between the algebra and its predual, and the spectral norm, defined with the trace. We show that these two norms are equivalent but generally not isometric for $p \neq 2$, even in the associative case of nonabelian von Neumann algebras when viewed through the Jordan product, thereby answering an open question raised by the first author in a previous paper. We further analyze the geometry of these spaces in concrete examples as complex spin factors or the complexified Albert algebra. Finally, we discuss the relevance of these results to generalized probabilistic theories (GPTs), where Jordan structures arise naturally, and explain why $\mathrm{JBW}$-algebras and their preduals provide a natural framework for such models.