C*-algebras
E286298
C*-algebras are a class of norm-closed, self-adjoint operator algebras on Hilbert spaces that form a fundamental framework in functional analysis and noncommutative geometry.
All labels observed (1)
| Label | Occurrences |
|---|---|
| C*-algebras canonical | 6 |
Statements (54)
| Predicate | Object |
|---|---|
| instanceOf |
Banach *-algebra
ⓘ
mathematical structure ⓘ operator algebra ⓘ |
| baseField | complex numbers ℂ ⓘ |
| characterizedBy |
Gelfand–Naimark theorem
ⓘ
representation as norm-closed *-subalgebras of B(H) ⓘ |
| containsExample |
AF-algebras
ⓘ
B(H), the algebra of all bounded operators on a Hilbert space H ⓘ C(X), the algebra of continuous complex-valued functions on a compact Hausdorff space X ⓘ Cuntz algebras ⓘ UHF-algebras ⓘ commutative C*-algebras of continuous functions ⓘ group C*-algebras ⓘ matrix algebras M_n(ℂ) ⓘ |
| definedOn | complex vector space ⓘ |
| field |
functional analysis
ⓘ
mathematical physics ⓘ noncommutative geometry ⓘ operator theory ⓘ |
| hasProperty |
closed under adjoint operation
ⓘ
closed under operator norm ⓘ complete with respect to its norm ⓘ norm-closed ⓘ self-adjoint ⓘ |
| hasStructure |
associative algebra
ⓘ
involution * ⓘ norm ⓘ |
| hasSubclass |
commutative C*-algebras
ⓘ
non-unital C*-algebras ⓘ nuclear C*-algebras ⓘ separable C*-algebras ⓘ simple C*-algebras ⓘ unital C*-algebras ⓘ von Neumann algebras (as special C*-algebras with extra structure) ⓘ |
| historicalPeriod | developed in the 1940s ⓘ |
| notation | C*-algebra ⓘ |
| originatedBy |
Israel Gelfand
ⓘ
Mark Naimark ⓘ |
| relatedConcept |
Gelfand representation of commutative C*-algebras
ⓘ
surface form:
Gelfand duality
K-theory for C*-algebras ⓘ noncommutative topology ⓘ quantum mechanics observables ⓘ spectral theory ⓘ |
| satisfiesAxiom |
(a + b)* = a* + b*
ⓘ
(ab)* = b* a* ⓘ (λa)* = \/bar{λ} a* for λ in ℂ ⓘ C*-identity ||a||^2 = ||a* a|| ⓘ ||a*|| = ||a|| ⓘ |
| typicalRealization |
algebra of bounded operators on a Hilbert space
ⓘ
norm-closed *-subalgebra of B(H) ⓘ |
| usedIn |
classification of operator algebras
ⓘ
index theory ⓘ mathematical formulation of quantum physics ⓘ noncommutative geometry ⓘ
surface form:
noncommutative geometry of Alain Connes
|
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.