Kosyak A. Regular, quasiregular and induced representations of infinite-dimensional groups

Thesis for the Doctor degree is devoted to the unitary representation theory of infinite-dimensional groups. The analogue of notions of regular, quasi-regular and induced representations are introduced for a first time for a wide class of infinite-dimensional groups. Ismagilov's conjecture (and its generalization given by the writer of a thesis) explains when these representations could be irreducible. Ismagilov's conjecture is proved for some matrix groups and for some classe of measures. Whether Ismagilov's conjecture holds for general groups over the field of real or complex numbers is on open problem. It is shown that Ismagilov's conjecture does not hold for quasi-regular representations. The von Neumann algebra generated by the right or left regular representations are also investigated. For infinite-dimensional groups the conditions are found when the commutant of the von Neumann algebra generated by the right regular representation coincides with the von Neumann algebra generated by the left regular representation. We investigate also when the mentioned von Neumann algebra is factor and it is shown that this factor is of type III_1.