Smooth paths of conditional expectations

Let A be a von Neumann algebra with a finite trace $τ$, represented in $H=L^2(A,τ)$, 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 $τ$-preserving conditional expectation. We say that the path $t\mapsto E_t$ is smooth if for every $a\in A$ and $v \in H$, the map $I\ni t\mapsto E_t(a)v\in H$ is continuously differentiable. This condition implies the existence of the derivative operator $dE_t(a):H\to H, \ dE_t(a)v=\frac{d}{dt}E_t(a)v.$ If this operator verifies the additional boundedness condition, $\int_J \|dE_t(a)\|_2^2 d t\le C_J\|a\|_2^2,$ 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, $G_t\circ E_0=E_t\circ G_t.$ 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.

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