Best approximants relative to a C${}^{\ast }$-subalgebra, joint numerical range and subdifferentials

We study the minimality of $n\times n$ Hermitian matrices $A$ respect to a $C^{\ast }$-subalgebra $\mathcal{B}$ of $M_n(\mathbb{C})$ in the spectral norm, that is $$\mathrm{\parallel }A\mathrm{\parallel }\le \mathrm{\parallel }A+B\mathrm{\parallel },\text{ for every }B\in \mathcal{B}\mathrm{.}$$ We generalize the notion of the moment of a subspace and relate it to the joint numerical range and the subdifferentials of the maximum eigenvalue. We extend results previously known for the subalgebra of diagonal operators and describe the subdifferential of the maximum eigenvalue in terms of the moment of the corresponding eigenspace. We also characterize $\mathcal{B}$-minimality via moments and subdifferentials, and provide examples.