Smooth paths of conditional expectations

Let A be a von Neumann algebra with a finite trace $\tau$, represented in $H=L^2(A,\tau )$, and let $B_t\subset A$ be sub-algebras, for $t$ in an interval $I$. Let $E_t:A\to B_t$ be the unique $\tau$-preserving conditional expectation. We say that the path $t\mapsto E_t$ is smooth if for every $a\in A$and$v \in H$,themap$$I\ni t\mapsto E_t(a)v\in H$$ is continuously differentiable. This condition implies the existence of the derivative operator $$d{E}_t(a):H\to H,d{E}_t(a)v=\frac{d}{dt}{E}_t(a)v\mathrm{.}$$ If this operator verifies the additional boundedness condition, $${\int }_J\mathrm{\parallel }d{E}_t(a){\mathrm{\parallel }}_2^{2}dt\le C_J\mathrm{\parallel }a{\mathrm{\parallel }}_2^2,$$ for any closed bounded sub-interval $J\subset I$, and $C_J>0$ a constant depending only on $J$, then the algebras $B_t$ are *-isomorphic. More precisely, there exists a curve $G_t:A\to A$, $t\in I$ of unital, *-preserving linear isomorphisms which intertwine the expectations, $$G_t\circ E_0=E_t\circ G_t\mathrm{.}$$ The curve $G_t$ is weakly continuously differentiable. Moreover, the intertwining property in particular implies that $G_t$ maps $B_0$ onto $B_t$. We show that this restriction is a multiplicative isomorphism.