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Let $M$ be a type III$_1$ factor and $\rho$ be a faithful normal state on $M$. If $p$ is a projection in $M$, can we find another normal state $\rho'$ on $M$ such that $\rho'(p)=0$ and $\|\rho-\rho'\|=k\rho(p)$ for some $k＞2$.
Let $M$ be a type III$_1$ factor and $\rho$ be a faithful normal state on $M$. If $p$ is a projection in $M$, can we find another normal state $\rho'$ on $M$ such that $\rho'(p)=0$ and $\|\rho-\rho'\|=k\rho(p)$ for some $k＞2$.
Let $M$ be a type III$_1$ factor and $\rho$ be a normal state on $M$. If $p$ is a projection in $M$, can we find another normal state $\rho'$ on $M$ such that $\rho'(p)=0$ and $\|\rho-\rho'\|=k\rho(p)$ for some $k＞2$.
Let $M$ be a type III$_1$ factor and $\rho$ be a faithful normal state on $M$. If $p$ is a projection in $M$, can we find another faithful normal normal state $\rho'$ on $M$ such that $\rho'(p)=0$ and $\|\rho-\rho'\|=k\rho(p)$ for some $k＞2$.
Let $M$ be a type III$_1$ factor and $\rho$ be a faithful normal state on $M$. If $p$ is a projection in $M$, can we find another faithful normal state $\rho'$ on $M$ such that $\rho'(p)=0$ and $\|\rho-\rho'\|=k\rho(p)$ for some $k＞2$.
Let $M$ be a type III$_1$ factor and $\rho$ be a faithful normal state on $M$. If $p$ is a projection in $M$, can we find another normal state $\rho'$ on $M$ such that $\rho'(p)=0$ and $\|\rho-\rho'\|=k\rho(p)$ for some $k＞2$.
