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arxiv: 2606.05456 · v1 · pith:P2C6KNULnew · submitted 2026-06-03 · 🧮 math.LO · math.OA

The computable functional calculus

Christopher J. Eagle , Timothy H. McNicholl This is my paper

Pith reviewed 2026-06-28 02:39 UTC · model grok-4.3

classification 🧮 math.LO math.OA
keywords computable functional calculusC*-algebrascomputable normal elementsspectrumspectral theoremcomputabilityoperator algebras
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The pith

The continuous functional calculus is a computable map on normal elements of computably presented C*-algebras.

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

The paper establishes that if a C*-algebra is given by a computable presentation and a normal element is given by a computable sequence of rational combinations, then the map sending each continuous function on the spectrum to the corresponding operator is itself computable. This means approximations to f(a) can be produced algorithmically whenever f is continuous and suitably presented. A reader cares because the result turns several classical existence statements in operator algebras into effective constructions, including compactness of spectra and approximate units.

Core claim

The continuous functional calculus is computable: given a computably presented C*-algebra, a computable normal element a, and a continuous function f on the spectrum of a (presented via a computable sequence of polynomials or similar), one can computably produce a sequence of elements converging to f(a) in the algebra norm.

What carries the argument

The continuous functional calculus map, which sends a continuous scalar function on the spectrum to the corresponding element of the C*-algebra generated by the normal element.

If this is right

  • The spectrum of any computable normal element is computably compact.
  • Every computably presented C*-algebra admits an effective approximate unit.
  • There is an effective version of the spectral theorem for compact operators on separable Hilbert spaces.

Where Pith is reading between the lines

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

  • The result supplies a uniform algorithmic way to build new computable operators from old ones via continuous functions, which may streamline constructions in effective operator theory.
  • It suggests that other theorems relying on the functional calculus, such as certain perturbation results, could admit effective versions under the same hypotheses.
  • The computability of the functional calculus may interact with computable presentations of Hilbert spaces to yield effective diagonalization procedures for normal operators.

Load-bearing premise

The notions of computably presented C*-algebra and computable normal element are taken as given and already strong enough to make the functional calculus map computable.

What would settle it

A specific computably presented C*-algebra together with a computable normal element a and a simple continuous function such as f(z) = |z| for which no algorithm produces a computable sequence converging to |a|.

read the original abstract

We show that the continuous functional calculus is computable. As consequences we obtain the computable compactness of the spectrum of any computable normal element of a computably presented $\mathrm{C}^*$-algebra, the existence of effective approximate units for computably presented $\mathrm{C}^*$-algebras, and an effective version of the Spectral Theorem for compact operators on separable Hilbert spaces.

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 / 1 minor

Summary. The manuscript claims to prove that the continuous functional calculus is computable once a C*-algebra is computably presented and the normal element is computable. Consequences include the computable compactness of the spectrum of any such normal element, the existence of effective approximate units for computably presented C*-algebras, and an effective version of the Spectral Theorem for compact operators on separable Hilbert spaces.

Significance. If the central claim holds, the result supplies a computable version of a core tool in operator algebras, directly yielding effective spectral compactness, approximate units, and a spectral theorem without free parameters or ad-hoc axioms. This strengthens the bridge between computability theory and C*-algebras and enables algorithmic access to classical results in separable Hilbert spaces.

minor comments (1)
  1. Abstract: the phrases 'computably presented C*-algebra' and 'computable normal element' are used without even a one-sentence gloss; a brief parenthetical definition or reference to the standard notions would improve immediate readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for noting its potential significance in bridging computability theory and C*-algebras. No major comments were listed in the report, so we provide no point-by-point responses below. We remain available to address any specific questions or concerns the referee may have upon request.

Circularity Check

0 steps flagged

No significant circularity; result is a theorem from given definitions

full rationale

The paper establishes that the continuous functional calculus is computable for computably presented C*-algebras and computable normal elements, yielding consequences like computable spectrum compactness. The central claim is a direct theorem whose proof relies on the provided definitions of computable presentation and computability of elements; these are taken as primitive inputs rather than derived from the result itself. No equations reduce a prediction to a fitted parameter by construction, no self-citation chain bears the load of the main claim, and no ansatz or renaming is smuggled in. The derivation is self-contained against external benchmarks in computable analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, invented entities, or non-standard axioms are visible in the abstract. The work rests on standard definitions from computable analysis and C*-algebra theory.

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discussion (0)

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Reference graph

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18 extracted references · 4 canonical work pages · 1 internal anchor

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