JBW*-代数的谱非结合L^p空间
Spectral nonassociative $\mathrm{L}^p$-spaces for $\mathrm{JBW}^*$-algebras
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中文总结 AI 辅助
该研究完成了一般JBW*-代数的迹谱非结合L^p空间构造，解决了JW*-代数对应结果的遗留例外情况，为相关Jordan代数模型提供了复Banach空间框架。
中文摘要 AI 辅助
我们完成了一般JBW*-代数的迹谱非结合L^p空间的构造。更确切地说，若M是配备正规有限忠实迹τ的JBW*-代数，且1≤p<∞，我们证明||x||_{L^p(M)}≜(τ[(x^*∘x)^(p/2)])^(1/p)（其中x∈M）定义了M上的范数。这解决了JW*-代数对应结果中遗留的剩余例外情况。主要难点在于复化Albert代数H₃(ℂ)，它无法嵌入结合算子代数。为处理该情况，我们建立了Kiefer映射Mₙ×Hₙ⁺⁺→Hₙ⁺（(a,h)↦a*h⁻¹a）联合凸性的Jordan类比，以及Carlen和Lieb变分公式的Jordan版本。这为一些广义概率理论中出现的Jordan代数模型提供了复Banach空间框架。
英文摘要
We complete the construction of tracial spectral nonassociative $\mathrm{L}^p$-spaces for general $\mathrm{JBW}^*$-algebras. More precisely, if $\mathcal{M}$ is a $\mathrm{JBW}^*$-algebra equipped with a normal finite faithful trace $τ$ and $1 \leq p < \infty$, we prove that $\|x\|_{\mathrm{L}^p(\mathcal{M})} \overset{\mathrm{def}}{=} (τ[(x^* \circ x)^{\frac p2}])^{\frac1p}$, where $x \in \mathcal{M}$, defines a norm on $\mathcal{M}$. This resolves the remaining exceptional case left open by the corresponding result for $\mathrm{JW}^*$-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 $\mathrm{M}_n \times \mathrm{H}_n^{++} \to \mathrm{H}_n^{+}$, $(a,h) \mapsto a^*h^{-1}a$, 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.
