  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 $v \in H$, 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.
