Suppose we have a $C^\ast$ algebra $A$ and a linear map $\phi\colon A\to A$, and assume that $\phi$ does not satisfy $\phi(x)^\ast\ne \phi(x^\ast)$ for some $x$, i.e. is not a $\ast$-homomorphism.
Question: Is there a name for the map $$\overline\phi\colon A\to A, \quad \overline{\phi}(x):=\phi(x^\ast)^\ast$$ which is linear and has the properties $\phi(x)^\ast=\overline\phi(x^\ast)$ and $\overline{\overline{\phi}}=\phi$?