A question on standard form in von Neumann algebra

Question

Let $M$ be a vN algebra (represented GNS space with respect to state) in standard form. Under which condition we can say a subalgebra $B$ of $M$ is also in standard form? If there exist $\varphi$ preserving conditional expectation $E_{\varphi}: M\rightarrow B$, will $B$ be in standard form (on GNS space of M)?

Answer 1

I'm not an expert on von Neumann algebras either, but here are some easy observations. 1. Having a conditional expectation is not sufficient. Any state on $M$ can be regarded as a conditional expectation onto the one dimensional subalgebra $B$ consisting of the scalar multiples of the identity. If ${\rm dim}(M) > 1$ then $L^2(M)$ will be too big to be the Hilbert space of the standard representation of this $B$. 2. $L^2(M)$ will never be naturally the same as $L^2(B)$, if $B$ is properly contained in $M$. 3. On the other hand, you could get lucky and have the representation of $B$ be unitarily equivalent to its standard representation. For instance, if $M = B(H)$ then its standard rep is as $B(H)\otimes I$ on $H \otimes H$. If $H = K\otimes K$ and $B = B(K)\otimes I \subset M$ then restricting the standard rep of $M$ to $B$ gives you $B(K)\otimes I\otimes I \otimes I$, which if $K$ is infinite dimensional is unitarily equivalent to $B(K)\otimes I$.