Pith. sign in

REVIEW 2 major objections 4 minor 1 cited by

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

T0 review · 2 major / 4 minor · reviewed 2026-08-27 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For every $\mathrm{JBW}^*$-algebra with a normal finite faithful trace, the spectral formula $\|x\|_{\mathrm{L}^p(\mathcal{M})}=(\tau[(x^*\circ x)^{p/2}])^{1/p}$ defines a norm, closing the exceptional Albert case.

desk verdict Completes the exceptional Albert-factor case for tracial spectral nonassociative Lp-spaces with a solid new proof; the general theorem's only real liability is its dependence on companion preprints that a referee must verify. read the letter →

arxiv 2608.14231 v1 pith:CFJIHCOA submitted 2026-08-14 math.FA math.OA

Cédric Arhancet This is my paper · ORCID
classification math.FAmath.OA MSC 46L5117C65
keywords nonassociativeLp-spacesJBW*-algebrasAlbertalgebraexceptionalJordantracialjointconvexitycomplexinterpolationspectralnorm
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that for every $\mathrm{JBW}^*$-algebra $\mathcal{M}$ with a normal finite faithful trace $\tau$ and $1\le p<\infty$, the formula $\|x\|_{\mathrm{L}^p(\mathcal{M})}=(\tau[(x^*\circ x)^{p/2}])^{1/p}$ defines a genuine norm on $\mathcal{M}$. The missing case was the complexified Albert algebra $\mathrm{H}_3(\mathbb{O}_{\mathbb{C}})$, which admits no embedding into an associative operator algebra. The proof establishes a Jordan analogue of the joint convexity of the map $(a,h)\mapsto a^*h^{-1}a$ and a variational formula for fractional powers, which together give the Minkowski inequality for $1\le p<2$. This completes the spectral construction for arbitrary tracial $\mathrm{JBW}^*$-algebras and supplies a complex Banach-space framework for Jordan-algebraic probabilistic models.

What carries the argument

The load-bearing mechanism is the variational formula for fractional powers on a finite-dimensional $\mathrm{JBW}$-algebra $A$ with trace $\tau$: $$\tau(u^q)=\inf_{h\in $A^{{++}}$}\{q\tau(u\circ $h^{{-1}}$)+(1-q)\tau($h^{{q/(1-q)}}$)\}, \quad 0<q<1,$$ together with the joint convexity of the Jordan quadratic-representation map $(a,h)\mapsto U_a(h^{-1})$ on $A\times A^{++}$. Joint convexity is proved through a unital positive Jordan map and the order inequality of [RoY82]; the variational formula is proved by spectral decomposition and an elementary scalar inequality. Together these ingredients make the $p$-th power of the candidate norm a convex, positively homogeneous function on $A\oplus A$, which yields the triangle inequality without any associative representation.

What would settle it

Take explicit matrices $x,y\in\mathrm{H}_3(\mathbb{O}_{\mathbb{C}})$ with octonion entries and compute, for a fixed $p\in[1,2)$ such as $p=1$, the traces $\tau[(x^*\circ x)^{p/2}]$, $\tau[(y^*\circ y)^{p/2}]$, and $\tau[((x+y)^*\circ(x+y))^{p/2}]$ using the finite-dimensional spectral decomposition. Since the formula is claimed to be a norm, any instance of $\|x+y\|_{\mathrm{L}^p}>\|x\|_{\mathrm{L}^p}+\|y\|_{\mathrm{L}^p}$ would refute Theorem 5.1; a computer search over rational-coefficient pairs would settle the triangle inequality.

Watch

Extended reading notes

Core claim

The central claim is Theorem 5.1: if $\mathcal{M}$ is a $\mathrm{JBW}^*$-algebra with a normal finite faithful trace and $1\le p<\infty$, then (1.5) is a norm. To reach it, the paper first treats the Albert factor $\mathrm{H}_3(\mathbb{O}_{\mathbb{C}})$ by writing $x=a+ib$ and identifying $\|x\|_{\mathrm{L}^p}$ with the square root of the real nonassociative $L^{p/2}$ norm of $a^2+b^2$ from [Ioc86]. For $p\ge 2$, an order estimate gives the triangle inequality; for $1\le p<2$, the paper proves joint convexity of the Jordan fractional map $(a,h)\mapsto U_a(h^{-1})$ and the identity $\tau(u^q)=\inf_{h\in A^{++}}\{q\tau(u\circ h^{-1})+(1-q)\tau(h^{q/(1-q)})\}$, then derives convexity of the $p$-th power of the candidate norm. It next shows that every normal finite faithful trace on $L^\infty(\Omega,\mathrm{H}_3(\mathbb{O}_{\mathbb{C}}))$ disintegrates fiberwise, making the spectral space isometric to a Bochner space. Combining this with the structural decomposition of a $\mathrm{JBW}$-algebra into special and purely exceptional parts yields the general theorem.

Load-bearing premise

The general theorem imports, without proof, the earlier norm result for tracial $\mathrm{JW}^*$-algebras and the conditional-expectation results used in Section 6; if any of those imported theorems are wrong, the general norm statement and the optimal comparison for the exceptional part would fail, even though Sections 3 and 4 are self-contained.

Editorial extensions

If this is right

  • For every tracial $\mathrm{JBW}^*$-algebra and $1\le p<\infty$, the completion of $\mathcal{M}$ under the spectral formula is a genuine complex Banach space; previously this was known only for $\mathrm{JW}^*$-algebras.
  • On $L^\infty(\Omega,\mathrm{H}_3(\mathbb{O}_{\mathbb{C}}))$ the spectral $\mathrm{L}^p$-norm is isometric to the Bochner space $L^p(\Omega,\nu,\mathrm{L}^p(\mathrm{H}_3(\mathbb{O}_{\mathbb{C}})))$.
  • The sharp comparison between spectral and interpolation norms, with universal constant $2^{|1/p-1/2|}$, remains optimal on the purely exceptional part, so the distortion does not grow there.
  • On selfadjoint elements the new norm reduces to the real nonassociative $\mathrm{L}^p$ norm of [Ioc86], so the complex theory is a faithful extension of the real one.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: the convexity machinery in Section 3 does not seem to rely on the exceptional factor's specific dimension, so a similar variational argument may prove the norm formula for other $\mathrm{JBW}^*$-algebras that lack an associative realization.
  • Editorial extension: the Jordan variational formula for $\tau(u^q)$, by analogy with the associative setting, could serve as a building block for R\'enyi-type divergences and hypothesis-testing quantities on Jordan probabilistic state spaces.
  • Editorial extension: Section 6 reduces every element to a finite-dimensional special subalgebra; this suggests a general transfer principle: any norm inequality that holds on all finite-dimensional $\mathrm{JW}^*$-subalgebras and is stable under the Bochner and interpolation constructions should extend automatically to the Albert factor.
Share X Bluesky LinkedIn Reddit HN

Formalized claims in Lean

  1. Claim #1: The central claim is Theorem 5.1: if $\mathcal{M}$ is a $\mathrm{JBW}^*$-algebra with a normal finite faithful trace and $1\le p<\infty$, then (1.5) is a norm. To reach it, the paper first treats the Albert factor $\mathrm{H}_3(\mathbb{O}_{\mathbb{C}})$ by writing $x=a+ib$ and identifying $\|x\|_{\mathrm{L}^p}$ with the square root of the real nonassociative $L^{p/2}$ norm of $a^2+b^2$ from [Ioc86

Signed reviews

No signed human review yet.

Request a human review

A listed scientist reviews the paper for a fee and the review publishes here regardless of verdict. See the reviewers or get listed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper proves that for any JBW*-algebra M equipped with a normal finite faithful trace τ and for 1 ≤ p < ∞, the expression ||x||_{L^p(M)} = (τ[(x*∘x)^{p/2}])^{1/p} defines a norm on M. The genuinely new part concerns the exceptional Albert factor H_3(O_C) and its measurable analogues L^∞(Ω,H_3(O_C)), which admit no embedding into an associative operator algebra. Section 3 establishes, for this factor, a Jordan analogue of the joint convexity of the matrix perspective and a Jordan Carlen–Lieb variational formula; these are used to prove the triangle inequality for all p in [1,∞). Section 4 shows that every normal finite faithful trace on L^∞(Ω,H_3(O_C)) disintegrates as the integral of the normalized fiber trace, giving a Bochner-space description of the resulting L^p-space. Section 5 combines the new exceptional result with the special (JW*) case, which is imported from the same-author preprint [ArL26, Theorem 3.7], to obtain the stated norm theorem for arbitrary JBW*-algebras. Section 6 compares the spectral norm with the complex-interpolation norm on the exceptional component, establishing the same optimal universal constants as in the JW* case.

Significance. If the imported theorem [ArL26, Theorem 3.7] is correct, the paper achieves a genuine completion: the spectral formula defines a complex Banach norm for every tracial JBW*-algebra, including the exceptional component for which no associative realization exists. The new methods are inventive and well executed: Proposition 3.4 gives a Jordan analogue of the joint convexity of the matrix perspective, Proposition 3.6 proves a variational formula that transfers scalar inequalities to the Albert factor through spectral decompositions, and Proposition 4.1 provides a clean trace-disintegration argument. The paper is mostly self-contained in its genuinely new parts, and the optimal comparison constants 2^{|1/p−1/2|} are an interesting addition. The main weakness is the reliance on self-cited preprints for the special summand, which leaves the central theorem conditional on an unverified external result.

major comments (2)
  1. [§5, Theorem 5.1] The proof of the norm theorem for arbitrary JBW*-algebras is not self-contained: after reducing to the direct sum A_sp ⊕ A_exp via the Shultz decomposition (5.1), the norm property on the special summand M_sp is justified only by the sentence "∥·∥_{L^p(M_sp)} is a norm by [ArL26, Theorem 3.7]". Since [ArL26] is an unpublished same-author preprint and its Theorem 3.7 is a load-bearing component of the theorem, an interested reader cannot verify the decisive part of the proof of Theorem 5.1. Please either include a complete proof of the special case in an appendix, or state explicitly in the abstract and introduction that the result is conditional on [ArL26, Theorem 3.7], and provide the full statement and a verification strategy. As it stands, the abstract's claim "We complete the construction" is stronger than what is actually proved in this manuscript.
  2. [§5, Theorem 5.1 and §6, Lemma 6.1] The proof of the triangle inequality for the exceptional factor also imports [ArL26, Proposition 3.1] for absolute homogeneity and point separation, and Lemma 6.1 of Section 6 imports [Arh24a, Propositions 2.5 and 3.11] for conditional expectations. These results are not proved or reproduced in the manuscript. While the comparison theorem in Section 6 is not the central claim, the conditional-expectation imports affect the sharpness proof of the constants. The authors should clarify the status of these preprints and, where possible, include the needed statements and proofs or provide published references.
minor comments (4)
  1. [Competing interests] The sentence "The authors declares that they have no competing interests" contains a grammatical error; since the paper has a single author, it should read "The author declares".
  2. [§4, Proposition 4.1] The notation "1_B 1" for the central projection in L^∞(Ω,H_3(O_C)) is slightly confusing; consider writing e_B = 1_B ⊗ 1 or explicitly defining it as the element with value 1 on B and 0 elsewhere.
  3. [§6, Theorem 6.3] The optimality proof relies on [ArL26, Remark 4.8] and [ArL26, Theorem 5.1] for the special subalgebra N_0; these are self-citations to a preprint and should be flagged as such in the text, so the reader knows the sharpness statement is also conditional on the same external results.
  4. [§3, Proposition 3.6] The proof uses "normalized" trace, but the argument never uses τ(1)=1; the assumption could be weakened to finite faithful trace without changing the proof.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the exceptional-case proof is self-contained and the self-citations are legitimate dependencies.

full rationale

The paper's derivation chain is not circular. The genuinely new claim for the exceptional Albert factor is proved directly: Proposition 3.1 reduces the spectral norm to Iochum's real L^q norm; Propositions 3.4 and 3.6 establish joint convexity and a Jordan Carlen–Lieb variational formula from the external Kadison inequality [RoY82] and a written spectral proof; Lemma 3.9 gives convexity; Theorem 3.10 then proves the triangle inequality for H3(OC) by distinguishing p ≥ 2 and 1 ≤ p < 2. Theorem 4.2 transfers this to L^∞(Ω,H3(OC)) through an explicit fiberwise trace disintegration and Bochner-space identification. The general Theorem 5.1 combines this exceptional part with the Shultz decomposition (5.1); the special summand is handled by citing [ArL26, Theorem 3.7]. That citation is same-author, but it is a previously stated theorem for the subclass of JW*-algebras, not a restatement of the present exceptional result, and it does not depend on this paper. Similarly, Section 6 uses [ArL26, Theorem 5.1] for special algebras and [Arh24a, Propositions 2.5 and 3.11] from a published source for conditional expectations. These are ordinary mathematical dependencies, not reductions by construction, self-definition, fitted parameters renamed as predictions, or imported uniqueness theorems. Any concern about the correctness or verification status of [ArL26] is a correctness risk, not circularity.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

Pure mathematical proof: no fitted parameters, no new entities. The paper's contribution is the exceptional Albert factor, but the general theorem imports several results from self-cited preprints ([ArL26], [Arh24a]) and classical structure theory. The main risk is the unverified status of these self-cited preprints.

assumptions (9)
  • standard math Kadison inequality for positive unital maps between JB-algebras ([RoY82, Theorem 1.2])
    Used in Proposition 3.4 to prove joint convexity of the Jordan perspective U_a(h^{-1}); published standard result.
  • standard math Spectral decomposition of positive selfadjoint elements in finite-dimensional JBW-algebras ([FaK94, Theorem III.1.1])
    Used in Proposition 3.6 and Lemma 3.9 to reduce scalar inequalities to finite spectra.
  • standard math Uniqueness of the normalized trace on the Albert algebra H_3(O) ([AlS03, Proposition 5.25])
    Used in Proposition 4.1 to show traces on L∞(Ω,H_3(O_C)) disintegrate as integrals of the fibre trace.
  • domain assumption Shultz structure theorem: a JBW-algebra splits into a special JW-algebra summand and a purely exceptional summand L∞_R(Ω,H_3(O)) ([Shu79, Theorem 3.9])
    Load-bearing for Theorem 5.1 and Corollary 6.4; it distributes the general case onto the special and exceptional components.
  • domain assumption For tracial JW*-algebras, the spectral formula (1.5) defines a norm ([ArL26, Theorem 3.7])
    Imported from the author's own preprint with L. Li; this result covers the special summand in Theorem 5.1 and is a load-bearing external input.
  • domain assumption Sharp two-sided comparison between spectral and interpolation norms for tracial JW*-algebras ([ArL26, Theorem 5.1])
    Imported in Section 6 and transferred to the exceptional factor via special subalgebras.
  • domain assumption Trace-preserving Jordan conditional expectations onto special subalgebras and their induced contractive projections ([Arh24a, Propositions 2.5 and 3.11])
    Used in Lemma 6.1 to compute both norms of an exceptional element inside a finite-dimensional special subalgebra.
  • standard math Shirshov-Cohn theorem: the Jordan algebra generated by two elements is special ([CGRP14, Theorem 3.1.55])
    Used in Lemma 6.1 to find the finite-dimensional special subalgebra N_x.
  • standard math Vector-valued complex interpolation theorem ([HvNVW16, Theorem 2.2.6])
    Used in Proposition 6.2 to identify L_{p,A}(L∞(Ω,H_3(O_C))) with a Bochner space.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Spectral nonassociative $\mathrm{L}^p$-spaces for $\mathrm{JBW}^*$-algebras." pith.science (2026). https://pith.science/paper/CFJIHCOA

@misc{pith2026260814231,
  author       = {Pith},
  title        = {Pith review of: Spectral nonassociative $\mathrmL^p$-spaces for $\mathrmJBW^*$-algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CFJIHCOA}},
  note         = {Machine review of arXiv:2608.14231}
}
abstract

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 $\tau$ and $1 \leq p < \infty$, we prove that $\|x\|_{\mathrm{L}^p(\mathcal{M})} \overset{\mathrm{def}}{=} (\tau[(x^* \circ x)^{\frac p2}])^{\frac1p}$, 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 to Jordan-algebraic models arising in some generalized probabilistic theories.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Sufficient positive maps between von Neumann algebras: R\'enyi divergences

    quant-ph 2026-08 accept novelty 6.0 of 10

    The authors extend recovery theorems for α-z Rényi divergences from 2-positive maps to merely positive maps between general von Neumann algebras using Jordan algebraic techniques.

Reference graph

Works this paper leans on

73 extracted references · 18 canonical work pages · cited by 1 Pith paper

  1. [1]

    A. A. Albert. On a certain algebra of quantum mechanics. Ann. of Math. (2) 35 (1934), no. 1, 65--73

    1934

  2. [2]

    E. M. Alfsen and F. W. Shultz. Geometry of state spaces of operator algebras. Mathematics: Theory & Applications. Birkhauser Boston, Inc., Boston, MA, 2003

    2003

  3. [3]

    Arhancet

    C. Arhancet. Nonassociative ^p -spaces and embeddings in noncommutative ^p -spaces. J. Math. Anal. Appl. 537 (2024), no. 1, Paper No. 128307, 24 pp

    2024

  4. [4]

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

    C. Arhancet and L. Li. Spectral versus interpolation norms in tracial nonassociative ^p -spaces. Preprint, arXiv:2604.23232

    work page Pith review arXiv

  5. [5]

    S. A. Ayupov. Extension of traces and type criterions for Jordan algebras of selfadjoint operators. Math. Z. 181 (1982), no. 2, 253--268

    1982

  6. [6]

    S. A. Ayupov and R. Z. Abdullaev. The Radon-Nikodym theorem for weights on semifinite JBW-algebras. Math. Z. 188 (1985), no. 4, 475--484

    1985

  7. [7]

    S. A. Ayupov. Center-valued traces on real operator algebras. Funktsional. Anal. i Prilozhen. 26 (1992), no. 2, 1--9, 96; translation in Funct. Anal. Appl. 26 (1992), no. 2, 77--83

    1992

  8. [8]

    S. A. Ayupov, A. Rakhimov and S. Usmanov. Jordan, real and Lie structures in operator algebras. Mathematics and its Applications, 418. Kluwer Academic Publishers Group, Dordrecht, 1997

    1997

Show all 73 references
  1. [9]

    J. Baez. The octonions. Bull. Amer. Math. Soc 39 (2002), 145--205

    2002

  2. [10]

    J. Barrett. Information processing in generalized probabilistic theories. Phys. Rev. A 75 (2007), 032304

    2007

  3. [11]

    Barnum, M

    H. Barnum, M. A. Graydon and A. Wilce. Some Nearly Quantum Theories. Proceedings QPL 2015

    2015

  4. [12]

    Barnum, M

    H. Barnum, M. A. Graydon and A. Wilce. Composites and Categories of Euclidean Jordan Algebras. Quantum 4, 359 (2020)

    2020

  5. [13]

    Barnum, C

    H. Barnum, C. Ududec and J. van de Wetering. Self-duality and Jordan structure of quantum theory follow from homogeneity and pure transitivity. Preprint, arXiv:2306.00362

    arXiv

  6. [14]

    Barnum and A

    H. Barnum and A. Wilce Information Processing in Convex Operational Theories. Electronic Notes in Theoretical Computer Science 270 (2011), no. 1, 3--15

    2011

  7. [15]

    Barnum and A

    H. Barnum and A. Wilce. Local Tomography and the Jordan Structure of Quantum Theory. Found Phys 44, 192--212 (2014)

    2014

  8. [16]

    Bergh and J

    J. Bergh and J. L\"ofstr\"om. Interpolation spaces. An Introduction. Springer-Verlag, Berlin, Heidelberg, New York, 1976

    1976

  9. [17]

    Boyd and L

    S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, Cambridge, 2004

    2004

  10. [18]

    Cabrera Garcia and A

    M. Cabrera Garcia and A. Rodriguez Palacios. Non-associative normed algebras. Vol. 1. The Vidav-Palmer and Gelfand-Naimark theorems. Encyclopedia of Mathematics and its Applications, 154. Cambridge University Press, Cambridge, 2014

    2014

  11. [19]

    Cabrera Garcia and A

    M. Cabrera Garcia and A. Rodriguez Palacios. Non-associative normed algebras. Vol. 2. Representation theory and the Zel'manov approach. Encyclopedia of Mathematics and its Applications, 167. Cambridge University Press, Cambridge, 2018

    2018

  12. [20]

    E. A. Carlen and E. H. Lieb. A Minkowski type trace inequality and strong subadditivity of quantum entropy. II. Convexity and concavity. Lett. Math. Phys. 83 (2008), no. 2, 107--126

    2008

  13. [21]

    E. A. Carlen. Inequalities in Matrix Algebras. Graduate Studies in Mathematics, vol. 251. Amer. Math. Soc., Providence, RI, 2025

    2025

  14. [22]

    Chiribella, G

    G. Chiribella, G. M. D'Ariano, and P. Perinotti. Informational derivation of quantum theory. Phys. Rev. A 84 (2011), 012311

    2011

  15. [23]

    K. Cho. Effectuses in Categorical Quantum Foundations. PhD thesis, Radboud University, 2019

    2019

  16. [24]

    C.-H. Chu. Jordan structures in geometry and analysis. Cambridge Tracts in Mathematics, 190. Cambridge University Press, Cambridge, 2012

    2012

  17. [25]

    E. B. Davies and J. T. Lewis. An operational approach to quantum probability. Comm. Math. Phys. 17 (1970), 239--260

    1970

  18. [26]

    J. Dixmier. Formes lin\'eaires sur un anneau d'op\'erateurs. (French). Bull. Soc. Math. France 81 (1953), 9--39

    1953

  19. [27]

    C. M. Edwards. On Jordan W^* -algebras. Bull. Sci. Math. (2) 104 (1980), no. 4, 393--403

    1980

  20. [28]

    G. M. Escolano, A. M. Peralta and A. R. Villena. The Mackey-Gleason-Bunce-Wright problem for vector-valued measures on projections in a ^* -algebra. Preprint, arXiv:2509.03213

    arXiv

  21. [29]

    Faraut and A

    J. Faraut and A. Kor\'anyi. Analysis on symmetric cones. Oxford Math. Monogr. Oxford Sci. Publ. The Clarendon Press, Oxford University Press, New York, 1994

    1994

  22. [30]

    Farenick, S

    D. Farenick, S. Jaques, and M. Rahaman. The fidelity of density operators in an operator-algebraic framework. J. Math. Phys. 57 (2016), no. 10, 102202

    2016

  23. [31]

    R. Furber. Categorical Duality in Probability and Quantum Foundations. PhD thesis, Radboud University, 2016

    2016

  24. [32]

    Hamhalter, O

    J. Hamhalter, O. Kalenda and A. M. Peralta. Determinants in Jordan matrix algebras. Linear Multilinear Algebra 71 (2020), no. 6, 961--1002

    2020

  25. [33]

    Hanche-Olsen and E

    H. Hanche-Olsen and E. St rmer. Jordan operator algebras. Monographs and Studies in Mathematics, 21. Pitman (Advanced Publishing Program), Boston, MA, 1984

    1984

  26. [34]

    Hardy and W

    L. Hardy and W. K. Wootters. Limited Holism and Real-Vector-Space Quantum Theory. Found Phys 42 (2012), 454--473

    2012

  27. [35]

    L. Hardy. Quantum Theory From Five Reasonable Axioms. Preprint, arXiv:quant-ph/0101012

    arXiv

  28. [36]

    Hyt\"onen, J

    T. Hyt\"onen, J. van Neerven, M. Veraar and L. Weis. Analysis in Banach spaces, Volume I: Martingales and Littlewood-Paley theory. Springer, 2016

    2016

  29. [37]

    B. Iochum. C\^ones autopolaires et alg\`ebres de Jordan. (French) [Self-dual cones and Jordan algebras]. Lecture Notes in Mathematics, 1049. Springer-Verlag, Berlin, 1984

    1984

  30. [38]

    B. Iochum. Nonassociative L^p -spaces. Pacific J. Math. 122 (1986), no. 2, 417--433

    1986

  31. [39]

    J. M. Isidro. Jordan triple systems in complex and functional analysis. Mathematical Surveys and Monographs, 243. American Mathematical Society, Providence, RI, 2019

    2019

  32. [40]

    Janotta and H

    P. Janotta and H. Hinrichsen. Generalized probability theories: what determines the structure of quantum theory?. J. Phys. A: Math. Theor. 47 (2014), no. 32, 323001

    2014

  33. [41]

    Uber die Multiplikation quantenmechanischer Gr\

    P. Jordan. \"Uber die Multiplikation quantenmechanischer Gr\"o ßen. Z. Physik 80 (1933), 285--291

    1933

  34. [42]

    Jordan, J

    P. Jordan, J. von Neumann and E. Wigner. On an algebraic generalization of the quantum mechanical formalism. Ann. of Math. (2) 35 (1934), no. 1, 29--64

    1934

  35. [43]

    J. Kiefer. Optimum experimental designs. J. Roy. Statist. Soc. Ser. B 21 (1959), 272--310

    1959

  36. [44]

    W. P. C. King. Semifinite traces on -algebras. Math. Proc. Cambridge Philos. Soc. 93 (1983), no. 3, 503--509

    1983

  37. [45]

    L. Lami. Non-classical correlations in quantum mechanics and beyond. PhD thesis, Universitat Aut\` o noma de Barcelona, 2017

    2017

  38. [46]

    L. Lami, C. Palazuelos, and A. Winter. Ultimate data hiding in quantum mechanics and beyond. Comm. Math. Phys. 361 (2018), 661--708

    2018

  39. [47]

    L. Lami, B. Regula, R. Takagi, and G. Ferrari. Framework for resource quantification in infinite-dimensional general probabilistic theories. Phys. Rev. A 103 (2021), Paper No. 032424

    2021

  40. [48]

    G. Ludwig. Foundations of Quantum Mechanics I. Theoretical and Mathematical Physics. Springer-Verlag, Berlin Heidelberg, 1983

    1983

  41. [49]

    G. Ludwig. An Axiomatic Basis for Quantum Mechanics: Volume 1, Derivation of H ilbert Space Structure. Springer-Verlag, Berlin Heidelberg, 1985

    1985

  42. [50]

    van Luijk and H

    L. van Luijk and H. Wilming. Sufficiency and Petz recovery for positive maps. Preprint, arXiv:2604.08380

    arXiv

  43. [51]

    van Luijk, A

    L. van Luijk, A. Marrakchi, T. J. Osborne, A. Stottmeister, and H. Wilming. Quantum steering is equivalent to state-preserving conditional expectations. Preprint, arXiv:2608.10783

    arXiv

  44. [52]

    McCrimmon

    K. McCrimmon. Jordan algebras and thneir applications. Bull. Am. Math. Soc. 844 (1978), 612--627

    1978

  45. [53]

    McCrimmon

    K. McCrimmon. A taste of Jordan algebras. Universitext. Springer-Verlag, New York, 2004

    2004

  46. [54]

    R. E. Megginson. An introduction to Banach space theory. Graduate Texts in Mathematics, 183. Springer-Verlag, New York, 1998

    1998

  47. [55]

    M. P. M\"uller. Probabilistic theories and reconstructions of quantum theory. SciPost Phys. Lect. Notes 28 (2021), 1--41

    2021

  48. [56]

    Niestegge

    G. Niestegge. Local tomography and the role of the complex numbers in quantum mechanics. Proc. A (2020) 476 (2238): 20200063

    2020

  49. [57]

    G. K. Pedersen and E. St rmer. Traces on Jordan algebras. Canadian J. Math. 34 (1982), no. 2, 370--373

    1982

  50. [58]

    Pl\'avala

    M. Pl\'avala. General probabilistic theories: An introduction. Physics Reports, Volume 1033, 7 September 2023, Pages 1--64

    2023

  51. [59]

    Renou, D

    M.-O. Renou, D. Trillo, M. Weilenmann, T. P. Le, A. Tavakoli, N. Gisin, A. Acin and M. Navascues. Quantum theory based on real numbers can be experimentally falsified. Nature 600 (2021), 625--629

    2021

  52. [60]

    A. G. Robertson and M. A. Youngson. Positive projections with contractive complements on Jordan algebras. J. Lond. Math. Soc. (2) 25 (1982), 365--374

    1982

  53. [61]

    M. B. Ruskai. Yet another proof of the joint convexity of relative entropy. Lett. Math. Phys. 112 (2022), art. 81

    2022

  54. [62]

    Shahandeh

    F. Shahandeh. Contextuality of general probabilistic theories. PRX Quantum 2 (2021), 010330

    2021

  55. [63]

    F. W. Shultz. On normed Jordan algebras which are Banach dual spaces. J. Funct. Anal. 31 (1979), no. 3, 360--376

    1979

  56. [64]

    Sonoda, H

    K. Sonoda, H. Arai and M. Hayashi. Hypothesis testing and Stein's lemma in general probability theories with Euclidean Jordan algebra and its quantum realization. Preprint, arXiv:2505.02487

    arXiv

  57. [65]

    J. M. Steele. The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities. MAA Problem Books Series. Math. Assoc. America and Cambridge Univ. Press, Cambridge, 2004

    2004

  58. [66]

    E. St rmer. Jordan algebras of type I. Acta Math. 115 (1966), 165--184

    1966

  59. [67]

    D. M. Topping. Jordan algebras of self-adjoint operators. Mem. Amer. Math. Soc. 53 (1965)

    1965

  60. [68]

    H. Upmeier. Symmetric Banach manifolds and Jordan C^* -algebras. North Holland, Amsterdam, 1985

    1985

  61. [69]

    H. Upmeier. Jordan Algebras in Analysis, Operator Theory, and Quantum Mechanics. Regional Conference Series in Mathematics No. 67, Amer. Math. Soc., Providence, 1987

    1987

  62. [70]

    Wang and Z

    S. Wang and Z. Wang. Operator means in JB-algebras. Rep. Math. Phys. 88 (2021), no. 3, 383--398

    2021

  63. [71]

    A. Wilce. Conjugates, Filters and Quantum Mechanics. Quantum 3, 158 (2019)

    2019

  64. [72]

    A. Wilce. Generalized Probability Theory: notes for a short course. Preprint, arXiv:2501.00718

    arXiv

  65. [73]

    V. J. Wright and S. Weigert. General probabilistic theories with a Gleason-type theorem. Quantum 5 (2021), 588

    2021

Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.