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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

$C^*$ -algebra

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Summary

A complex associative Banach $*$ -algebra satisfying the $C^*$ -identity $\lVert a^*a\rVert=\lVert a\rVert^2$ .

Record metadata

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

Carrier(s)

Banach *-algebra $A$

Axioms / constraints

$\lVert a^*a\rVert=\lVert a\rVert^2$

Notes

Some authors build the unit into “ $C^*$ -algebra”; others allow nonunital $C^*$ -algebras. This atlas does not require a unit at this node.

Concept sources

Wikipedia — $C^*$-algebra — C*-algebra · expository · source ID wp-cstar-algebra
nLab — $C^*$-algebra — C*-algebra · reference · source ID nlab-cstar-algebra
Princeton — Functional Analysis Lecture Notes — Functional Analysis Lecture Notes · course notes · source ID princeton-functional-analysis

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Banach algebra + involution(construction junction) $C^*$ -algebra

Permalink to relation

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

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

Authored explanation

Impose the $C^*$ -identity $\lVert a^*a\rVert=\lVert a\rVert^2$ on the compatible Banach $*$ -algebra.

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

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

$C^*$ -algebra Quantum observable

Permalink to relation

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

Relation ID: cstar_to_observable
Relation type: Mathematical formulation mathematical-formulation
Direction: source → target
Endpoint roles: source: Mathematically formulates; target: Mathematically formulated using
Authored annotation: observable algebra

Authored explanation

Operator-algebraic quantum theory organizes bounded observables in $C^*$ -algebras.

How to interpret this relation type

A mathematical concept supplies part of the formal language, state space, representation, or analytic machinery used by a scientific or mathematical-physics concept. The source is the mathematical predecessor; this does not claim that physical content follows from mathematics alone.

Relation sources

Wikipedia — Observable — Observable · encyclopedic reference · source ID wp-physics-observable
nLab — quantum observable — quantum observable · mathematical physics reference · source ID nlab-physics-quantum_observable
Wikipedia — $C^*$-algebra — C*-algebra · expository · source ID wp-cstar-algebra
nLab — $C^*$-algebra — C*-algebra · reference · source ID nlab-cstar-algebra

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$C^*$ -algebra

Summary

Record metadata

Carrier(s)

Axioms / constraints

Notes

Concept sources

Incoming relations (arrows to this concept)

Banach algebra + involution(construction junction) $C^*$ -algebra

Authored explanation

How to interpret this relation type

Relation sources

Outgoing relations (arrows from this concept)

$C^*$ -algebra Quantum observable

Authored explanation

How to interpret this relation type

Relation sources