Let $M$ be a type III$\_1$ factor. Suppose $\rho$ is a normal state on $M$, given any $c\in [0,2]$, can we find a normal state $\rho'$ on $M$ such that $\|\rho-\rho'\|=c$? Or can we find a sequence of normal states $\rho\_n$ such that $\|\rho-\rho\_n\|\to c$ as $n\to \infty$?



Normal states on a type III$_1$ factor
I don't think you can get $c = 2$, but you can do this for any $c \in [0,2)$. Given $\epsilon > 0$, find a projection $p \in M$ with $\rho(p) < \epsilon$ and then find $\rho'$ supported on $p$. Then $$(\rho - \rho')(1 - 2p) = 1 - 2\rho(p) - 1 + 2 = 2(1 - \rho(p)),$$ showing that $\|\rho - \rho'\|$ can be arbitrarily close to $2$. Then linearly interpolating between $\rho$ and $\rho'$ gives you normal states of any intermediate distance to $\rho$.


