  We consider a group $GL(\infty)$ and its parabolic subgroup $B$ corresponding
to partition $\infty=\infty+m+\infty$. Denote by $P$ the kernel of the natural
homomorphism $B\to GL(m)$. We show that the space of double cosets of
$GL(\infty)$ by $P$ admits a natural structure of a semigroup. In fact this
semigroup acts in subspaces of $P$-fixed vectors of some unitary
representations of $GL(\infty)$ over finite field.
