L-p-improving Convolution Operators on Finite Quantum Groups

We characterize positive convolution operators on a finite quantum group G that are L-p-improving. More precisely, we prove that the convolution operator T-phi : x -> phi * x given by a state phi on C (G) satisfies there exists 1 < p < 2, vertical bar vertical bar T-phi : L-p (G) -> L-2 (G) vertical bar vertical bar = 1 if and only if the Fourier series phi satisfies vertical bar vertical bar phi (alpha) vertical bar vertical bar < 1 for all nontrivial irreducible unitary representations a, and if and only if the state (phi o S) * phi is non-degenerate (where S is the antipode). We also prove that these Lp-improving properties are stable under taking free products, which gives a method to construct Lp-improving multipliers on infinite compact quantum groups. Our methods for non-degenerate states yield a general formula for computing idempotent states associated with Hopf images, a formula that generalizes earlier work of Banica, Franz, and Skalski.