Factor-representation of the group $GL(\infty)$ and admissible representations $GL(\infty)^X$

The paper is the first of three parts of the work which studies factor-representations of III-type of $ GL(\infty)$ group. Let $\mathfrak A$ be a complex finite-dimensional algebra with unit ${\mathbf 1}_{\mathfrak A}$, let $G(\mathfrak A ) $ designate a group of all infinite dimensional invertible matrices with values on $\mathfrak A$. The complete classification of unitary representations of $G(\mathfrak A )$, which are spherical with respect to unitary subgroup $U(\infty)\subset GL(\infty)=G(\mathbb{C}{\mathbf 1}_{\mathfrak A})\subset G(\mathfrak A)$, was obtained in the work. To each representation there corresponds a class of factor-representations $\Pi$ of $GL(\infty)$ group with the property, that there exists nonzero vector $\xi$ in a space of the representation $H_{\Pi}$, which suffices to correlation: $\varphi(g)=(\Pi(g)\xi,\xi)=\varphi(ugu^*)$ for all $u\in U(\infty)$. We give a complete description of representations which satisfy the last condition in further parts of the work.