Definition of a centralizer of a weight

Question

The following definition is from Takesaki's book, Vol2, Chapter 8, Definition 2.1.

Let $\varphi$ be a faithful weight in a von Neumann algebra $M$. Set $M_{\varphi}=\{x\in M:\sigma_{\varphi}^t(x)=x,t\in \Bbb R\}$. We call $M_{\varphi}$ the centralizer of $\varphi$.

What is the explicit expression of $\sigma_{\varphi}^t(x)$ in the above definition?

Answer 1

Here $$ \sigma_\varphi^t(x)=\Delta_\varphi^{it} x \Delta_\varphi^{-it}, $$ where $\Delta_\varphi$ is the modular operator of the GNS representation of $\varphi$.

I have to say that if you are reading Tomita Theory and you have to ask what the modular group is, you are reading way ahead of where you should be reading.