I had to prove some statements for two unital $C^*$ algebras $A, B$ and $\Phi: A \to B$ a surjective $*$-homomorphism. I started out the proof by stating
By the rank-nullity theorem, $A$ and $B$ are isomorphic as vector spaces. Because $\Phi$ respects the algebra operations, they are also isomorphic as $C^*$ algebras.
After this, the rest of the proof followed (very) easily. However, when checking my solution, I saw that it followed a much more difficult path. Therefore I doublechecked, but I do not seem to find any mistake, In hindsight it does feel a little 'too easy'.
Am I right in claiming the above?
The rank-nullity theorem only holds in the finite-dimensional case and so I can not apply it here. See the comments for counterexamples.