$L_p$-improving convolution operators on finite quantum groups

Abstract

We characterize positive convolution operators on a finite quantum group $\mathbb{G}$ which are $L_p$-improving. More precisely, we prove that the convolution operator $T_{\phi }:x\mapsto \phi \star x$ given by a state $\phi$ on $C(\mathbb{G})$ satisfies \[ \exists1<p<2,\quad\|T_φ:L_{p}(\mathbb{G})\to L_{2}(\mathbb{G})\|=1 \] if and only if the Fourier series $\hat{\phi }$ satisfy $\Vert \hat{\phi }(\alpha )\Vert <1$ for all nontrivial irreducible unitary representations $\alpha$, if and only if the state $(\phi \circ S)\star \phi$ is non-degenerate (where $S$ is the antipode). We also prove that these $L_p$-improving properties are stable under taking free products, which gives a method to construct $L_p$-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.

20 pages. Final version. Minor corrections

References in corpus (2)

• The Fourier transform in quantum group theory
• Fourier transform on locally compact quantum groups

Cited by in corpus (5)

• The Ergodic Theorem for Random Walks on Finite Quantum Groups
• Tracing the orbitals of the quantum permutation group
• Isolated partial Hadamard matrices, and related topics
• Diaconis-Shahshahani Upper Bound Lemma for Finite Quantum Groups
• Modelling questions for transitive quantum groups