Image of $C^\ast$-algebra is closed?

Question

Let $A$ be a non-zero commutative $C^\ast$ algebra and let $\varphi : A \to B$ be a homomorphism of star algebras.

Please could someone help me how to show that $\varphi(A)$ is closed in $B$?

Answer 1

The fact that $\varphi $ is a $*$-homomorphism implies that $\bar\varphi:A/\ker\varphi\to\varphi (A) $, given by $\bar\varphi (a+\ker\varphi)=\varphi (a) $, is isometric and onto.

So any Cauchy sequence in $\varphi (A) $ comes from a Cauchy sequence in $ A/\ker\varphi $. As the latter is closed (in general the quotient of a Banach space by a Banach subspace is closed), so is $\varphi (A) $.

Edit: suppose that $\varphi:A\to B$ is isometric. Let $\{b_n\}\subset\varphi(A)$ be Cauchy. We have $b_n=\varphi(a_n)$ for appropriate $a_n\in A$. Since $$\|b_n-b_m\|=\|\varphi(a_n)-\varphi(a_m)\|=\|\varphi(a_n-a_m)\|=\|a_n-a_m\|,$$ we deduce that $\{a_n\}$ is Cauchy. With $A$ complete, there exists $a\in A$ with $a=\lim a_n$. As $\varphi$ is continuous, $\varphi(a)=\lim\varphi(a_n)=\lim b_n$. So $\varphi(A)$ is complete.