Flip automorphism for a $II_1$ factor is not inner

Question

It is known that for a $II_1$ factor $M$, the flip automorphism defined on $M \overline{\otimes} M$ by $a \otimes b \mapsto b \otimes a$ is not inner. A proof can be found on Vol. IV of the books by Kadison & Ringrose. I just found that in this MathOverflow post the accepted answer by Jesse Peterson had provided a simple proof for this result. However I don't understand why the inequalities in the inner product calculations are true. Could someone provide a more detailed explanation? Thank you very much.

Answer 1

The key inequality is $|\tau(x)|\leq\tau(|x|)$. I cannot really follow what Jesse is doing in his first inequality, but all we need to do it take the triangle inequality to get a sum of terms $$ |(\tau\otimes\tau)((p_k\otimes1)(v^*\otimes w^*)U(p_k\otimes 1))|. $$ Then, with $x=(p_k\otimes1)(v^*\otimes w^*)U(p_k\otimes 1)$, \begin{align} x^*x&=(p_k\otimes1)U^*(v\otimes w)(p_k\otimes1)(v^*\otimes w^*)U(p_k\otimes 1) =(p_k\otimes 1)U^*(vp_kv^*\otimes1)U(p_k\otimes1)\\ &=p_k\otimes vp_kv^*. \end{align} As this is a projection, $|x|=(x^*x)^{1/2}=x^*x=p_k\otimes vp_kv^*$. So \begin{align} |(\tau\otimes\tau)((p_k\otimes1)(v^*\otimes w^*)U(p_k\otimes 1))| &=|(\tau\otimes\tau)(x)|\leq(\tau\otimes\tau)(|x|)=(\tau\otimes\tau)(p_k\otimes vp_kv^*)\\ &=\tau(p_k)^2. \end{align}