Spectral nonassociative L^p-spaces for JBW^*-algebras

Abstract

We complete the construction of tracial spectral nonassociative L^p-spaces for general JBW^*-algebras. More precisely, if ℳ is a JBW^*-algebra equipped with a normal finite faithful trace τ and 1 ≤ p < ∞, we prove that |x|_L^p(ℳ) oversetdef= (τ[(x^* ∘ x)^p/2])^1/p, where x ∈ ℳ, defines a norm on ℳ. This resolves the remaining exceptional case left open by the corresponding result for JW^*-algebras. The main difficulty is the complexified Albert algebra H₃(𝕆_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 M_n × H_n⁺⁺ → H_n⁺, (a,h) ↦ a^*h⁻¹a, where 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.

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BibTeX RIS

BibTeX

@misc{arhancet2026spectral,
  title = {{Spectral nonassociative $\mathrmL^p$-spaces for $\mathrmJBW^*$-algebras}},
  author = {Cédric Arhancet},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.14231},
  eprint = {2608.14231},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.14231},
}

RIS

TY  - UNPB
TI  - Spectral nonassociative L^p-spaces for JBW^*-algebras
AU  - Cédric Arhancet
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2608.14231
UR  - https://arxiv.org/abs/2608.14231
AB  - We complete the construction of tracial spectral nonassociative L^p-spaces for general JBW^*-algebras. More precisely, if ℳ is a JBW^*-algebra equipped with a normal finite faithful trace τ and 1 ≤ p < ∞, we prove that |x|_L^p(ℳ) oversetdef= (τ[(x^* ∘ x)^p/2])^1/p, where x ∈ ℳ, defines a norm on ℳ. This resolves the remaining exceptional case left open by the corresponding result for JW^*-algebras. The main difficulty is the complexified Albert algebra H_3(𝕆_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 M_n × H_n^++ → H_n^+, (a,h) ↦ a^*h^−1a, where 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.
ER  -

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