Spectral nonassociative ${\mathrm{L}}^p$-spaces for ${\mathrm{JBW}}^{\ast }$-algebras

Abstract

We complete the construction of tracial spectral nonassociative ${\mathrm{L}}^p$-spaces for general ${\mathrm{JBW}}^{\ast }$-algebras. More precisely, if $\mathcal{M}$ is a ${\mathrm{JBW}}^{\ast }$-algebra equipped with a normal finite faithful trace $\tau$ and $1\le p<\infty$, we prove that $\Vert x{\Vert }_{{\mathrm{L}}^p(\mathcal{M})}\stackrel{\mathrm{def}}{=}(\tau [(x^{\ast }\circ x{)}^{\frac{p}{2}}]{)}^{\frac{1}{p}}$, where $x\in \mathcal{M}$, defines a norm on $\mathcal{M}$. This resolves the remaining exceptional case left open by the corresponding result for ${\mathrm{JW}}^{\ast }$-algebras. The main difficulty is the complexified Albert algebra ${\mathrm{H}}_3({\mathbb{O}}_{\mathbb{C}})$, which admits no embedding into an associative operator algebra. To treat this case, we establish a Jordan analogue of the joint convexity of the Kiefer map ${\mathrm{M}}_n\times {\mathrm{H}}_n^{++}\to {\mathrm{H}}_n^{+}$, $(a,h)\mapsto a^{\ast }{h}^{-1}a$, where ${\mathrm{H}}_n$ is the space of Hermitian matrices, and a Jordan version of a variational formula of Carlen and Lieb. This provides a complex Banach space framework for Jordan-algebraic models arising in some generalized probabilistic theories.

arXiv comment25 pages, minor improvements