I am reading a note written by Claire Anantharaman and Sorin Popa recently. Here is a link to the notes.
https://www.math.ucla.edu/~popa/Books/IIun.pdf
I am reading Chapter 13 right now, in section 13.1.2, they try to build a connection between a completely positive normal map from $M$ to $N$ and the bimodule $_MH_N$. I can understand most of their discussion, but I do have a question here.
In page 217, they claim that if we have a bimodule $_MH_N$ and a left $N$-bounded vector $\xi_0$, then we can construct completely positive normal map from $M$ to $N$ by using the definition $\phi:M\to N$, $\phi(x)=L_{\xi_0}^*\pi_M(x)L_{\xi_0}$, However, By the definition, we just know that $\phi(x)\in B(L^2(N))$. That's where I get confused, actually. I am not sure how to show that $\phi(x)\in N$ for every $x\in M$?
Any help will be truly grateful!
By definition, $L_{\xi_0}$ is a right module map, that is, $L_{\xi_0}(\eta y)=L_{\xi_0}(\eta)y$ for $y\in M$ and $\eta\in L^2(N)$. Thus $L_{\xi_0}^\ast$ is also a right module map. Moreover, $\pi_M(x)$ is a right module map by the definition of a bimodule. Hence $\phi(x)$ is a right module map on $L^2(N)$. But since $N$ is in standard form on $L^2(N)$, the bounded right module maps on $L^2(N)$ coincide with $N$.