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Construction junctions are diamonds. They show where multiple structures must coexist on the same carrier and satisfy compatibility conditions.

Canonical static concept record

*-algebra

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Summary

A complex associative algebra equipped with a conjugate-linear involutive anti-automorphism.

Record metadata

Canonical name: *-algebra
Concept ID: star_algebra
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Algebra
Domains: Algebra, Analysis
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

complex associative algebra $A$

Data

map $a\mapsto a^*$

Axioms / constraints

$(a+b)^*=a^*+b^*$
$(\lambda a)^*=\overline{\lambda}a^*$
$(ab)^*=b^*a^*$
$(a^*)^*=a$

Notes

The scalar involution here is complex conjugation.

Concept sources

Wikipedia — *-algebra — *-algebra · expository · source ID wp-star-algebra
nLab — *-algebra — *-algebra · reference · source ID nlab-star-algebra
Mathlib — The star operation, bundled as a star-linear equivalence — The star operation, bundled as a star-linear equivalence · formal reference · source ID mathlib-star-module

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Associative algebra *-algebra

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: e_assoc_star
Relation type: Add data add-data
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + involution

Authored explanation

Over a scalar field with involution, add a semilinear involutive anti-automorphism.

How to interpret this relation type

Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.

Relation sources

Wikipedia — *-algebra — *-algebra · expository · source ID wp-star-algebra
nLab — *-algebra — *-algebra · reference · source ID nlab-star-algebra
Stacks Project — Commutative Algebra (PDF) — Commutative Algebra (PDF) · reference text · source ID stacks-algebra-text

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

*-algebra Banach algebra + involution(construction junction)

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: j_cstar__star_algebra
Relation type: Combine compatible structures combine-compatible
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: involution

Authored explanation

Supply the conjugate-linear involutive anti-automorphism on the same complex algebra carrier used by the Banach-algebra input.

How to interpret this relation type

Feed several structures into a construction junction and impose compatibility between them.

Relation sources

Mathlib — Classes of $C^*$-algebras — Classes of C*-algebras · formal reference · source ID mathlib-cstar-classes

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*-algebra

Summary

Record metadata

Carrier(s)

Data

Axioms / constraints

Notes

Concept sources

Incoming relations (arrows to this concept)

Associative algebra *-algebra

Authored explanation

How to interpret this relation type

Relation sources

Outgoing relations (arrows from this concept)

*-algebra Banach algebra + involution(construction junction)

Authored explanation

How to interpret this relation type

Relation sources