-algebra homomorphisms have closed images
Theorem - Let be a *-homomorphism between the -algebras (http://planetmath.org/CAlgebra) and . Then has closed (http://planetmath.org/ClosedSet) image (http://planetmath.org/Function), i.e. is closed in .
Thus, the image is a -subalgebra of .
Proof: The kernel of , , is a closed two-sided ideal of , since is continuous (see this entry (http://planetmath.org/HomomorphismsOfCAlgebrasAreContinuous)). Factoring threw the quotient -algebra we obtain an injective *-homomorphism .
Injective *-homomorphisms between -algebras are known to be isometric (see this entry (http://planetmath.org/InjectiveCAlgebraHomomorphismIsIsometric)), hence the image is closed in .
Since the images and coincide we conclude that is closed in .
|Title||-algebra homomorphisms have closed images|
|Date of creation||2013-03-22 17:44:37|
|Last modified on||2013-03-22 17:44:37|
|Last modified by||asteroid (17536)|
|Synonym||image of -homomorphism is a -algebra|