Distance and best approximations in operator norm and trace class norm

We study the best approximation and distance problems in the operator space $\text{B}(\text{HS})$ and in the space of trace class operators ${\text{LS}}^1(\text{B}(\text{HS}))$. Formulations of distances are obtained in both cases. The case of finite-dimensional $C^{\ast }$-algebras is also considered. The computational advantage of the results is illustrated through examples.