Entropy characteristics of subsets of states. I

We study the properties of quantum entropy and $\chi$-capacity (regarded as a function of sets of quantum states) in the infinite-dimensional case. We obtain conditions for the boundedness and continuity of the restriction of the entropy to a subset of quantum states, as well as conditions for the existence of the state with maximal entropy in certain subsets. The notion of $\chi$-capacity is considered for an arbitrary subset of states. The existence of an optimal average is proved for an arbitrary subset with finite $\chi$-capacity. We obtain a sufficient condition for the existence of an optimal measure and prove a generalized maximal distance property.