Entropy, von Neumann and the von Neumann Entropy

The highway of the development of entropy is marked by many great names, for example, Clausius, Gibbs, Boltzmann, Szilard, von Neumann, Shannon, Jaynes, and several others. In this article the emphasis is put on von Neumann and on quantum mechanics. The selection of the subjects reflects the taste (and the knowledge) of the author and it must be rather restrictive. In the past 50 years entropy has broken out of thermodynamics and statistical mechanics and invaded communication theory, ergodic theory and shown up in mathematical statistics, social and life sciences. It is practically impossible to present all of its features. The favourite subjects of entropy is about macroscopic phenomena, irreversibility and incomplete knowledge. In the strictly mathematical sense entropy is related to the asymptotics of probabilities or it is a kind of asymptotic behaviour of probabilities.