$C^{*}$-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 $ℬ$.

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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 $ℬ$. $\mathrm{\square }$