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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?

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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 by Ozawa gives a quick introduction and some pointers to the literature (especially the work of Pisier).

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  • $\begingroup$ Thanks! That seems to be a good answer, and gives me other things to look at. $\endgroup$
    – Edwin Beggs
    Commented Jun 17, 2013 at 14:34

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