Semigroup methods in the representation theory of the infinite symmetric group

This thesis is devoted to giving a classification of all the unitary representations of the infinite symmetric group, which consists of bijections on natural numbers, that are continuous with respect to a suitable topology in the infinite symmetric group. For the first time in 1972 this classification was done by A. Lieberman without the use of semigroups . We work out the semigroup approach to this problem developed by G. I. Olshansky. In particular, we associate a sequence of representations of finite inverse symmetric semigroups to any unitary representation of the infinite symmetric group.