$L_p$-improving convolution operators on finite quantum groups

Abstract

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.

Isolated Partial Hadamard Matrices and Related Topics

“…In this section we briefly discuss isolation questions for the quantum algebraic objects constructed in section 5. In the square matrix case some probabilistic technology is available from [34], and for the Hadamard representations, the result is as follows: Theorem 6.1. The p-th moment of spectral measure of the quantum group G ⊂ S + N associated to an Hadamard matrix H ∈ M N (C) is the dimension of the 1-eigenspace of (T p ) i 1 ...ip,j 1 ...jp = tr(P i 1 j 1 .…”

Quantum groups, from a functional analysis perspective

“…Assume indeed that X is a probability space. We have then the following result, from [11], [48]: where r G = (ϕ • π) * r , with ϕ = tr ⊗ X being the random matrix trace. Proof.…”

Diaconis–Shahshahani Upper Bound Lemma for Finite Quantum Groups

“…The following definition is similar to that of Simeng Wang ((2.5), [24]) save for a choice of left-right. As remarked upon by Simeng Wang, his definition is similar to earlier definitions of Kahng and also Caspers save for the presence of the conjugate representation κ α rather than κ α itself.…”