We characterize positive convolution operators on a finite quantum group G which are Lp-improving. More precisely, we prove that the convolution operator Tϕ : x → ϕ ⋆ x given by a state ϕ on C(G) satisfiesif and only if the Fourier seriesφ satisfy φ(α) < 1 for all nontrivial irreducible unitary representations α, and if and only if the state (ϕ • S) ⋆ ϕ 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 to Hopf images, which generalizes earlier work of Banica, Franz and Skalski.

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Abstract. This paper is devoted to the study of Sidon sets, Λ(p)-sets and some related notions for compact quantum groups. We establish several different characterizations of Sidon sets, and in particular prove that any Sidon set in a discrete group is a strong Sidon set in the sense of Picardello. We give several relations between Sidon sets, Λ(p)-sets and lacunarities for L pFourier multipliers, generalizing a previous work by Blendek and Michalicek. We also prove the existence of Λ(p)-sets for orthogonal systems in noncommutative L p -spaces, and deduce the corresponding properties for compact quantum groups. Central Sidon sets are also discussed, and it turns out that the compact quantum groups with the same fusion rules and the same dimension functions have identical central Sidon sets. Several examples are also included.