DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Unital star algebra and positive normalized state

Definition

A unital complex star algebra is a complex vector space $A$ with a complex-bilinear associative product $(a, b) \mapsto a b$ , a specified unit $1_{A}$ , and a conjugate-linear map $a \mapsto a^{*}$ such that, for all $a, b \in A$ ,

$(a b)^{*} = b^{*} a^{*}, (a^{*})^{*} = a, 1_{A}^{*} = 1_{A} .$

A positive normalized state on $A$ is a complex-linear functional $ω : A \to C$ such that $ω (1_{A}) = 1$ and $ω (a^{*} a)$ is a real nonnegative number for every $a \in A$ . This is an algebraic definition: no norm, topology, boundedness condition, or completion is part of the data.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Unital star algebra and positive normalized state

Definition

Dependency tree · 0 levels

Sources