Take a (unital) algebra map $f:A\to B$ between two unital C\* algebras - not necessarily star preserving. Under what circumstances is there a $b\in B$ so that $g(a)=b\ f(a)\ b^{-1}$ is a star algebra map? If not, is there another method of modifying $f$ to get a star algebra map?



when is an algebra map conjugate to a star algebra map
I will assume "algebra map" means "homomorphism." Certainly a necessary condition is that $f$ is bounded. When $B=B(H)$ this question (is every bounded homomorphism similar to a \*-homomorphism?) is known as the "Kadison Similarity Problem" and is still open. The answer is known to be affirmative when $A$ is a nuclear C\*-algebra, and in a few other cases. I am not an expert in this area, but it is my impression that the answer is believed to be "no" in general. [This survey](http://www.kurims.kyoto-u.ac.jp/~narutaka/notes/similarity.pdf) by Ozawa gives a quick introduction and some pointers to the literature (especially the work of Pisier). 


