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. (C) 2018 Elsevier Inc. All rights reserved.