The $K$-functOr (Grothndieck group) of the infinite symmetric group.

The Grothendieck group $K_0(\sigma_\infty)$ of the group $\sigma_\infty$ of finite permutations of a countable set is described. We also discribe all semifinite characters of this group and use them to determine the cone of true $K_+^0(\sigma_\infty)$ representations.