  For an algebraic compact quantum group $H$ we establish a bijection between
the set of right coideal $*$-subalgebras $A\to H$ and that of left module
quotient $*$-coalgebras $H\to C$. It turns out that the inclusion $A\to H$
always splits as a map of right $A$-modules and right $H$-comodules, and the
resulting expectation $E:H\to A$ is positive (and lifts to a positive map on
the full $C^*$ completion on $H$) if and only if $A$ is invariant under the
squared antipode of $H$.
  The proof proceeds by Tannaka-reconstructing the coalgebra $C$ corresponding
to $A\to H$ by means of a fiber functor from $H$-equivariant $A$-modules to
Hilbert spaces, while the characterization of those $A\to H$ which admit
positive expectations makes use of a Fourier transform turning elements of $H$
into functions on $C$.
