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For an algebraic compact quantum group $H$ we establish a bijection betweenthe set of right coideal $*$-subalgebras $A\to H$ and that of left modulequotient $*$-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 theresulting expectation $E:H\to A$ is positive (and lifts to a positive map onthe full $C^*$ completion on $H$) if and only if $A$ is invariant under thesquared antipode of $H$.  The proof proceeds by Tannaka-reconstructing the coalgebra $C$ correspondingto $A\to H$ by means of a fiber functor from $H$-equivariant $A$-modules toHilbert spaces, while the characterization of those $A\to H$ which admitpositive expectations makes use of a Fourier transform turning elements of $H$into functions on $C$.

Relative Fourier transforms and expectations on coideal subalgebras