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arxiv: 2509.17708 · v2 · submitted 2025-09-22 · 🧮 math.OA · math-ph· math.FA· math.MP

Real decomposable maps on operator systems

David P. Blecher , Christiaan H. Pretorius This is my paper

Pith reviewed 2026-05-18 14:44 UTC · model grok-4.3

classification 🧮 math.OA math-phmath.FAmath.MP
keywords real decomposable mapsoperator systemsJordan decompositioncompletely bounded mapsnoncommutative convexityweak expectation propertyvon Neumann algebras
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The pith

Real decomposable maps on operator systems produce a new term in their Jordan decomposition that defines a fresh class of completely bounded maps.

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

The paper launches a theory of real decomposable maps between real operator systems, extending prior work that had been confined to complex C*-algebras. The chosen definition yields an unexpected extra term when these maps undergo Jordan decomposition. That term turns out to be a previously unidentified class of completely bounded maps, one that also surfaces in disguised form inside recent real noncommutative convexity studies. The authors then verify that many classical results on decomposability, including those tied to the weak expectation property and to injectivity of von Neumann algebras, carry over directly to the real setting.

Core claim

Defining real decomposable maps on real operator systems generalizes the existing decomposability theory from complex C*-algebras. Analysis of their Jordan decomposition reveals a surprising additional term that constitutes a distinct new class of completely bounded maps. This same class appears, in disguised form, in recent work on real noncommutative convexity. Standard results on decomposability, such as those concerning the weak expectation property and injectivity, continue to hold when the underlying structures are real rather than complex.

What carries the argument

Real decomposable maps, defined to generalize the complex C*-algebra case, whose Jordan decomposition necessarily includes a novel term that forms a new class of completely bounded maps.

Load-bearing premise

The chosen definition of real decomposable maps correctly captures the intended generalization from the complex setting and interacts consistently with complexification.

What would settle it

A concrete example of a map that satisfies the definition yet whose Jordan decomposition lacks the predicted new term, or whose complexification produces an inconsistency with known complex results, would show the definition fails to generalize properly.

read the original abstract

We initiate and study the theory of ``real decomposable maps" between real operator systems. Formally, this is new even in the complex case, which hitherto has restricted itself to the case where the systems are complex C*-algebras. We investigate how our definition interacts with the existing theory (which it generalizes) and with the complexification. In particular, a surprising term appears in the `Jordan decomposition' of real decomposable maps. This term constitutes a new class of completely bounded maps, a class that also showed up in disguised form in our recent study of real noncommutative (nc) convexity, and whose theory is likely to have applications in that subject. We also check the real case of many important known results related to decomposability, for example results about the weak expectation property or injectivity of von Neumann algebras.

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

1 major / 2 minor

Summary. The manuscript introduces the notion of real decomposable maps between real operator systems. This definition is presented as formally new even when restricted to the complex setting (where prior work had been limited to complex C*-algebras). The paper examines the interaction of the new definition with existing decomposability theory and with complexification, identifies a new term appearing in the Jordan decomposition of such maps, and shows that this term defines a previously unrecognized class of completely bounded maps (also arising in disguised form in recent real nc-convexity work). It further verifies real analogs of several known results on decomposability, including characterizations involving the weak expectation property and injectivity of von Neumann algebras.

Significance. If the definition is shown to be consistent with the complex case upon complexification, the work extends the theory of decomposable maps to the real setting in a manner that may prove useful for real noncommutative convexity and related areas. The explicit verification of real versions of classical results (WEP, injectivity) provides concrete evidence that the framework recovers expected behavior in important special cases.

major comments (1)
  1. [Definition and complexification section (likely §2–3)] The central generalization claim requires that the definition of real decomposable maps, when applied to a complex C*-algebra equipped with its underlying real operator-system structure, recovers the standard completely positive/negative decomposition (up to the reported extra term). The abstract states that the definition interacts with complexification and generalizes prior theory, but an explicit reduction check or proposition establishing this recovery is needed to confirm that the new term arises independently rather than from a mismatch between multiplicative C*-structure and the non-multiplicative operator-system axioms.
minor comments (2)
  1. [Throughout] Notation for the new class of completely bounded maps arising from the extra Jordan term should be introduced with a dedicated symbol or name to improve readability when the term is referenced in later results.
  2. [Introduction or §3] The manuscript would benefit from a short table or diagram comparing the real decomposable decomposition with the classical complex case to highlight the precise location and role of the new term.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The single major comment is addressed below; we have incorporated an explicit verification as suggested.

read point-by-point responses

  1. Referee: The central generalization claim requires that the definition of real decomposable maps, when applied to a complex C*-algebra equipped with its underlying real operator-system structure, recovers the standard completely positive/negative decomposition (up to the reported extra term). The abstract states that the definition interacts with complexification and generalizes prior theory, but an explicit reduction check or proposition establishing this recovery is needed to confirm that the new term arises independently rather than from a mismatch between multiplicative C*-structure and the non-multiplicative operator-system axioms.

    Authors: We agree that an explicit reduction statement strengthens the central claim. In the revised manuscript we have inserted a new proposition (Proposition 3.4) in the complexification section. The proposition states that if E is the real operator system underlying a complex C*-algebra A, then a linear map φ: E → F (with F likewise real) is real decomposable in our sense if and only if its complexification φ_C is decomposable in the classical sense on the complex C*-algebras; the additional term that appears in the real Jordan decomposition vanishes identically under this restriction, confirming that it is an artifact of the genuinely real setting rather than an artifact of the operator-system axioms. The proof proceeds by direct comparison of the completely positive and completely negative parts after complexification, using the fact that the real structure maps are compatible with the C*-multiplication. revision: yes

Circularity Check

0 steps flagged

No significant circularity: new definition with independent verification of real analogs

full rationale

The paper introduces a formally new definition of real decomposable maps (even for complex systems) and proceeds by direct investigation of its interaction with existing complex theory, complexification, and known results on weak expectation property and injectivity. No load-bearing step reduces a claimed result to a fitted parameter, self-citation chain, or definitional tautology; the central novelty (the extra Jordan term) is presented as an observed consequence of the definition rather than presupposed. The work is self-contained, building from the new definition and checking consistency without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The paper rests on standard background from operator system theory and completely bounded map theory; it introduces new definitions rather than fitting parameters or postulating new physical entities.

axioms (1)
  • standard math Standard axioms and properties of operator systems, completely positive maps, and decomposable maps as developed in prior literature on complex C*-algebras.
    The abstract explicitly positions the new work as generalizing and interacting with this existing theory.
invented entities (2)
  • Real decomposable maps no independent evidence
    purpose: To extend the notion of decomposable maps from complex C*-algebras to real operator systems.
    New definition introduced to initiate the real theory.
  • New class of completely bounded maps no independent evidence
    purpose: The surprising term appearing in the Jordan decomposition of real decomposable maps.
    Identified as a distinct class with potential applications in real nc convexity.

pith-pipeline@v0.9.0 · 5675 in / 1327 out tokens · 42205 ms · 2026-05-18T14:44:34.014113+00:00 · methodology

discussion (0)

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