KMS-states on group $GL(\infty)$ and admissible representations $GL(\infty)^X$

The paper is the third and the last part of the work in which the complete description of factor-representations of group $GL(\infty)$, which are defined by unitary invariant positive-definite functions, is obtained. The necessary and sufficient conditions for KMS-states on $GL(\infty)$ group to turn the corresponding spherical representation of group $GL(\infty)\times GL(\infty)$ to standart one are given.