DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Complex associative algebra with involution

Definition

A complex associative algebra with involution is a unital associative algebra over $C$ as in Algebras over a commutative ring, central structure maps, and algebra homomorphisms, equipped with a conjugate-linear map $a \mapsto a^{*}$ using complex conjugation from Real and imaginary parts, complex conjugation, and modulus such that $(a^{*})^{*} = a$ , $(a b)^{*} = b^{*} a^{*}$ , and $1^{*} = 1$ . Its self-adjoint part is

$A_{s a} : = {a \in A : a^{*} = a} .$

The theorem The symmetrized product of an associative algebra is Jordan ↗ proves that this subset is a real vector subspace closed under the symmetrized product.

Depends on

Used by

The self-adjoint part need not be an associative subalgebra Counterexample
The symmetrized product of an associative algebra is Jordan Theorem

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

N. Jacobson, Lectures on Quadratic Jordan Algebras, §1 (standard reference, not scraped)
Holger P. Petersson, Albert Algebras, §4.3 (standard reference, not scraped)