  We initiate the effective study of conditional expectations onto subfactors.
Our main result is that if $M$ is an existentially closed II$_1$ factor with a
w-spectral gap subfactor $N$, then the conditional expectation function onto
$N$ can be computed from a (Turing) oracle that computes a presentation of $M$,
the inclusion of $N$ into $M$, and a spectral gap function for the pair
$(M,N)$.
