$C^{*}$-algebra homomorphisms have closed images

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

Thus, the image $f(𝒜)$ is a $C^{*}$-subalgebra of $ℬ$.

Proof: The kernel of $f$, $\mathrm{Ker}f$, is a closed two-sided ideal of $𝒜$, since $f$ is continuous (see this entry (http://planetmath.org/HomomorphismsOfCAlgebrasAreContinuous)). Factoring threw the quotient $C^{*}$-algebra $𝒜/\mathrm{Ker}f$ we obtain an injective *-homomorphism $\tilde{f}:𝒜/\mathrm{Ker}f⟶ℬ$.

Injective *-homomorphisms between $C^{*}$-algebras are known to be isometric (see this entry (http://planetmath.org/InjectiveCAlgebraHomomorphismIsIsometric)), hence the image $\tilde{f}(𝒜/\mathrm{Ker}f)$ is closed in $ℬ$.

Since the images $\tilde{f}(𝒜/\mathrm{Ker}f)$ and $f(𝒜)$ coincide we conclude that $f(𝒜)$ is closed in $ℬ$. $\mathrm{\square }$

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