C*-algebra homomorphisms have closed images

Theorem - Let f:𝒜 be a *-homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath between the C*-algebras (http://planetmath.org/CAlgebra) 𝒜 and . Then f has closed (http://planetmath.org/ClosedSet) image (http://planetmath.org/Function), i.e. f(𝒜) is closed in .

Thus, the image f(𝒜) is a C*-subalgebraMathworldPlanetmath of .

Proof: The kernel of f, Kerf, is a closed two-sided idealMathworldPlanetmath of 𝒜, since f is continuousPlanetmathPlanetmath (see this entry (http://planetmath.org/HomomorphismsOfCAlgebrasAreContinuous)). Factoring threw the quotient C*-algebra 𝒜/Kerf we obtain an injectivePlanetmathPlanetmath *-homomorphism f~:𝒜/Kerf.

Injective *-homomorphisms between C*-algebras are known to be isometric (see this entry (http://planetmath.org/InjectiveCAlgebraHomomorphismIsIsometric)), hence the image f~(𝒜/Kerf) is closed in .

Since the images f~(𝒜/Kerf) and f(𝒜) coincide we conclude that f(𝒜) is closed in .

Title C*-algebra homomorphisms have closed images
Canonical name CalgebraHomomorphismsHaveClosedImages
Date of creation 2013-03-22 17:44:37
Last modified on 2013-03-22 17:44:37
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 9
Author asteroid (17536)
Entry type Theorem
Classification msc 46L05
Synonym image of C*-homomorphism is a C*-algebra