Let $M$ be a vN algebra(represented GNS space with respect to state) is 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 will in standard form(On GNS space of M)? Please comment thanks in advance.

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)?

Let $M$ be a vN algebra(represented GNS space with respect to state) is in standard form, under which condition we can say a subalgebra $B$ of $M$ is also in standard form. Please comment thanks in advance.

Let $M$ be a vN algebra(represented GNS space with respect to state) is 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 will in standard form(On GNS space of M)? Please comment thanks in advance.