A Product on Double Cosets of $B_{\mathrm{\infty }}$

For some infinite-dimensional groups $G$ and suitable subgroups $K$ there exists a monoid structure on the set $K\mathrm{\backslash }G\mathrm{/}K$ of double cosets of $G$ with respect to $K$. In this paper we show that the group $B_{\mathrm{\infty }}$, of the braids with finitely many crossings on infinitely many strands, admits such a structure.