Let $A$ be a semisimple Banach algebra, $ x \mapsto x^{*} $ be an involution on $A$. Then it is continuous.
I've looked through some reference books and webpages, all of them prove this by using closed graph theorem, or the fact that every homomorphism from a Banach algebra onto a semisimple Banach algebra is continuous (some call this Johnson's Theorem) (eg. this post: Does *-operator be automatically continuous) However both of these theorems require the mapping under consideration to be linear, while an involution is anti-linear by definition; in particular $( \lambda x)^{*}=\bar{\lambda}x^{*} \neq \lambda x^{*}$ in general.
Am I missing something here as to why these methods of proof are right? Can someone supply me with a correct proof if the proofs are wrong? Thank you in advance!
You've got the point that there is some sloppiness when people deal with complex scalars. This case can be rectified however quite easily.
Apply any of these theorems to the complex conjugate of $A$ (which is still a semi-simple Banach algebra). Then the involution is linear on it.