Let $\rho_1$ and $\rho_2$ be two normal states on a von Neumann algebra $M$. We denote by $M_{\rho_1-\rho_2}$ the centralizer of $\rho_1-\rho_2$ and $M_{\rho_i}$ the centralizer of $\rho_i(i=1,2)$.
Does there exist relationship between $M_{\rho_1-\rho_2}$ and $M_{\rho_1}$? Under What condition,can we conclude that $M_{\rho_1-\rho_2}\subset M_{\rho_1}$?