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LIT-274: A new description of orthogonal bases

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Properties6
StatusActive — read in full 2026-09-27 (NOTE-247); worth reading as the short, self-contained source of the fact that "a commutative †-Frobenius algebra" and "an orthogonal basis" are the same data in FdHilb, and "special" the same as "orthonormal". The proof is complete in outline, and the heavy step is delegated to the spectral theorem for finite-dimensional commutative C*-algebras. Read §3's displayed formulas (3) and (4) as the orthonormal case only, and the result as complex and finite-dimensional only.
Version2
Tagsmathematics, quantum-foundations
Extended byLIT-275 — H*-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics
Cited as “paper” byNOTE-247 — A new description of orthogonal bases
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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Coecke, Pavlovic & Vicary (2008), Mathematical Structures in Computer Science 23(3), 555–567 (2013) — DOI-10.1017/S0960129512000047

Standing in the record

Filed on 2026-09-27 at the owner’s request, because THEORY-017’s promote_when names it: it is the result that makes “a basis” and “a commutative Frobenius algebra” one notion in that account. LIT-272 cites it too. published: is the arXiv v1 date (ADR-002).

It was filed Deferred, unread. NOTE-247 is the close reading of 2026-09-27, and it placed the work: Active — worth reading as the short, self-contained source of the fact that “a commutative †-Frobenius algebra” and “an orthogonal basis” are the same data in FdHilb, and “special” the same as “orthonormal”. The proof is complete in outline, and the heavy step is delegated to the spectral theorem for finite-dimensional commutative C*-algebras. Read §3’s displayed formulas (3) and (4) as the orthonormal case only, and the result as complex and finite-dimensional only.

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