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LIT-275: H*-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics

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StatusActive — read in full 2026-09-27 (NOTE-248); worth reading as the precise statement of how far "a basis is a Frobenius algebra" survives in infinite dimensions. It gives the unit's failure, the H*-algebra substitute and five equivalent conditions, all proved or reduced to Ambrose (1945), with one question left open. Read "arbitrary bases" in the abstract as "orthonormal bases of arbitrary dimension over ℂ", and "observables" as discrete-spectrum observables only.
Version2
Tagsmathematics, quantum-foundations
ExtendsLIT-274 — A new description of orthogonal bases
Cited as “paper” byNOTE-248 — H*-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics
Cited as “source” byTHEORY-017 — In a Hilbert-space model only unitarily invariant structure is intrinsic; a basis or tensor factorisation, and so any parts or features, is extra data supplied from outside the space, and the record's Hilbert-space works differ in where they get it, THEORY-721 — On a complex Hilbert space no orthonormal basis or tensor factorisation of given dimensions is distinguished by the space alone, and orthonormal bases correspond exactly to commutative special †-Frobenius algebras; over the reals that correspondence fails

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Abramsky & Heunen (2010), Mathematical Foundations of Information Flow (Clifford Lectures), AMS Proc. Symp. Applied Math. 71, 1–24 (2012) — DOI-10.1090/psapm/071/599

Standing in the record

Filed on 2026-09-27 at the owner’s request. It tests THEORY-017’s finite-dimensional boundary, and it extends LIT-274 (Coecke, Pavlovic & Vicary) beyond finite dimension. published: is the arXiv v1 date (ADR-002).

It was filed Deferred, unread. NOTE-248 is the close reading of 2026-09-27, and it placed the work: Active — worth reading as the precise statement of how far “a basis is a Frobenius algebra” survives in infinite dimensions. It gives the unit’s failure, the H*-algebra substitute and five equivalent conditions, all proved or reduced to Ambrose (1945), with one question left open. Read “arbitrary bases” in the abstract as “orthonormal bases of arbitrary dimension over ℂ”, and “observables” as discrete-spectrum observables only.

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