On the fixed point spaces of some completely positive maps

In this paper we generalize the results shown by Das and Peterson. Let $M$ be a ${\rm II}_1$-factor acting on $L^2(M)$. We consider certain unital normal completely positive maps on $B(L^2(M))$ which are identity on $M$. We investigate their fixed point spaces and obtain a rigidity result. As an application, we show some results of subfactors.