Dual norm only for a part of matrix

Question

A dual norm $\|\cdot\|^\circ$ of norm $\|\cdot\|$ can be given in terms of inner product $$\|A\|^\circ=\max_B |\text{Tr}(AB)|,$$ with the constraint $\|B\|\leq1$. This can be re-expressed, for unitarily invariant norms, as $$\|A\|^\circ=\max_B\|AB\|_1,$$ where $\|X\|_1=\text{Tr}(\sqrt{X^\dagger X})$ is the trace norm. Can this notion of dual norm can be extended to partial trace of bipartite operator? In other words, is there well developed results for the following maximization $$\max_B \|A(I\otimes B)\|_1,$$ where $\|B\| \leq1$? For example, $\|B\|$ can be Schatten-$p$ norm.