The Haagerup problem on subadditive weights on $W^*$-algebras. II

In year 1975 U. Haagerup has posed the following question: whether every normal subadditive weight on a $W^*$-algebra is $\sigma$-weakly lower semicontinuous? In year 2011 the author has positively answered this question in a particular case of abelian $W^*$-algebras and has presented a general form of normal subadditive weights on these algebras. Here we positively answer this question in the case of finite-dimensional $W^*$-algebras. As a corollary, we give a positive answer for subadditive weights with some natural additional condition on atomic $W^*$-algebras. We also obtain the general form of such normal subadditive weights and norms for wide class of normed solid spaces on atomic $W^*$-algebras.