Relative Fourier transforms and expectations on coideal subalgebras

For an algebraic compact quantum group H we establish a bijection between the set of right coideal ⁎-subalgebras A→H and that of left module quotient ⁎-coalgebras H→C. It turns out that the inclusion A→H always splits as a map of right A-modules and right H-comodules, and the resulting expectation E:H→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→H by means of a fiber functor from H-equivariant A-modules to Hilbert spaces, while the characterization of those A→H which admit positive expectations makes use of a Fourier transform turning elements of H into functions on C.