Nessonov N. Factor-representations of infinite dimensional matrix groups and invariant of operator algebras

Representations of infinite dimensional matrix groups. Full classifications of admissible representations and KMS-states on infinite dimensional classical groups. Methods of ergodic theory, representations theory, theory of operator algebras, methods of complex analysis, theory of symmetrical functions, probability theory. full description of spherical functions on infinite dimensional complex linear groups and group of motions is obtained, Full classification for the admissible representations of infinite dimensional classical groups is given, conception of KMS-state on group is introduced and full description of factor – representations of type III, that define by unitary invariant KMS-states is obtained, notation of Bogoliubov automorphism on representation of infinite dimensional matrix group is given and in the case of infinite dimensional unitary group explicit formula for its quantum entropy is obtained. The results and methods are important for applications in representations theory of infinite dimensional groups.