Oracle computability of conditional expectations onto subfactors

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 $(M,N)$.