Suppose $A^*$ is the adjoint of $A$, and $\|\cdot\|_D$ is the dual of the norm $\|\cdot\|$. When is the following true?
$$\|A\|_D=\|A^*\|$$
Two examples that come to mind are row-sum/column sum norms for matrix $A$ and operator norm/induced trace norm for positive linear map $T$ (Bhatia book, Equation 2.37)
Edit relevant section from Horn/Johnson "Matrix Analysis", page 357:
