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arxiv: 2604.23232 · v2 · submitted 2026-04-25 · 🧮 math.OA · math.FA· math.QA

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Spectral versus interpolation norms in tracial nonassociative L^p-spaces

C\'edric Arhancet, Lei Li

Pith reviewed 2026-05-08 07:02 UTC · model grok-4.3

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keywords nonassociative L^p spacestracial JW*-algebrasinterpolation normsspectral normsJordan algebrasvon Neumann algebrasgeneralized probabilistic theories
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The pith

The interpolation norm and spectral norm on tracial nonassociative L^p-spaces are equivalent but not isometric for p not equal to 2.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper compares two natural ways to equip nonassociative L^p-spaces with a norm when the underlying structure is a tracial JW*-algebra. One construction uses complex interpolation between the algebra and its predual; the other uses the spectral radius relative to the trace. The authors prove the resulting norms are equivalent, yet the unit balls differ except when p equals 2. The same non-isometry appears even if one begins with an ordinary nonabelian von Neumann algebra and replaces its product with the Jordan product. Concrete calculations are given for spin factors and the Albert algebra, and the metric distinction is linked to modeling choices in generalized probabilistic theories.

Core claim

We show that the interpolation norm arising from the complex method and the spectral norm defined with the trace are equivalent but generally not isometric for p ≠ 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 and discuss the relevance of these results to generalized probabilistic theories where Jordan structures arise naturally.

What carries the argument

The direct comparison of the complex interpolation norm (between a tracial JW*-algebra and its predual) with the trace-defined spectral norm on the associated nonassociative L^p-space.

If this is right

  • The two norms coincide isometrically if and only if p equals 2.
  • Non-isometry holds for the Jordan algebra obtained from any nonabelian von Neumann algebra.
  • Explicit ratios between the norms can be computed in finite-dimensional examples such as complex spin factors and the Albert algebra.
  • The choice of norm affects the geometry of the space when Jordan structures are used to model generalized probabilistic theories.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The non-isometry may produce different distance or probability measures depending on which norm is adopted in Jordan-based physical models.
  • The same norm distinction could appear in other nonassociative L^p constructions outside the tracial JW* setting.
  • Specifying the norm becomes necessary when Jordan preduals are employed as state spaces in generalized probabilistic theories.

Load-bearing premise

That the complex interpolation norm and the trace-based spectral norm are the two canonical definitions whose relationship must be settled in the tracial JW*-algebra setting.

What would settle it

An explicit element x in a concrete nonabelian von Neumann algebra equipped with the Jordan product, together with a value p ≠ 2, such that the interpolation norm of x exactly equals its spectral norm.

Figures

Figures reproduced from arXiv: 2604.23232 by C\'edric Arhancet, Lei Li.

Figure 1
Figure 1. Figure 1: Octonion multiplication table Consider the real Jordan algebra A def = H2(O) =  a x x b : a, b ∈ R, x ∈ O  , equipped with the Jordan product defined in (1.1). We introduce the following elements of A: I def =  1 0 0 1 , s1 def =  1 0 0 −1  , s2 def =  0 1 1 0 and sk+2 def =  0 ek −ek 0  for any integer k ∈ {1, . . . , 7}. Then A = RI ⊕ Rs1 ⊕ Rs2 ⊕ Rs3 ⊕ · · · ⊕ Rs9. Moreover, all these element… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript compares the complex interpolation norm (from the pair (A, A_*)) and the trace-induced spectral norm in tracial nonassociative L^p-spaces over JW*-algebras. It proves these norms are equivalent (induce the same topology) for 1 < p < ∞ but not isometric when p ≠ 2, including when a nonabelian von Neumann algebra is equipped only with its Jordan product; this resolves an open question from prior work by the first author. Concrete counterexamples are given in complex spin factors and the complexified Albert algebra, the geometry of the resulting spaces is analyzed, and implications for generalized probabilistic theories are discussed.

Significance. If the claims hold, the work is significant for clarifying the canonical norm on nonassociative L^p-spaces and for extending interpolation techniques to the Jordan setting without associativity assumptions. Strengths include the explicit, parameter-free counterexamples (complex spin factors, complexified Albert algebra) that demonstrate a p-dependent norm ratio, the adaptation of standard complex-method estimates that survive the Jordan identity, and the direct link to applications in GPTs where Jordan structures appear naturally.

minor comments (3)
  1. [§2] §2: the statement that the Jordan product is bounded with respect to the operator norm should be accompanied by an explicit reference to the relevant estimate in the cited literature on JBW-algebras.
  2. [§4.2] §4.2, Example 4.3 (complex spin factor): the explicit computation of the ratio between the two norms is given for specific p; a short remark on the behavior as p → 1^+ or p → ∞ would strengthen the geometric discussion.
  3. [§6] §6: the relevance to GPTs is sketched; adding one or two concrete references to standard texts on generalized probabilistic theories would improve accessibility for readers outside operator algebras.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive recommendation for minor revision. The provided summary accurately captures the main results: the equivalence of the interpolation and spectral norms on tracial nonassociative L^p-spaces for 1 < p < ∞, their failure to be isometric when p ≠ 2 (including in the associative case viewed through the Jordan product), the explicit counterexamples in complex spin factors and the complexified Albert algebra, and the discussion of implications for generalized probabilistic theories.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via independent proofs and examples

full rationale

The paper recalls the two canonical norms (complex interpolation between A and A_* versus trace-induced spectral norm) from standard prior literature on JW*-algebras, proves their equivalence using complex-method estimates that extend to the Jordan product without new assumptions, and establishes non-isometry by direct computation on explicit finite-dimensional examples (complex spin factors, complexified Albert algebra). The mention of an open question from the first author's earlier paper is purely contextual and does not supply any load-bearing step, uniqueness theorem, or fitted input; all central claims rest on explicit constructions, estimates, and counterexamples that are verifiable independently of the self-citation. No step reduces by construction to its own inputs or to a self-referential chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on established constructions from functional analysis and Jordan algebra theory without introducing new free parameters or postulated entities.

axioms (2)
  • standard math The complex interpolation method between a Banach space and its predual produces a family of norms.
    Standard tool in the theory of L^p spaces.
  • domain assumption Tracial JW*-algebras admit a spectral norm defined via the trace.
    Definition specific to the tracial Jordan setting of the paper.

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