Mehrdad Kalantar

POISSON BOUNDARIES OVER LOCALLY COMPACT QUANTUM GROUPS

We present versions of several classical results on harmonic functions and Poisson boundaries in the setting of locally compact quantum groups. In particular, the Choquet–Deny theorem holds for compact quantum groups; also, the result of Kaimanovich–Vershik and Rosenblatt, which characterizes group amenability in terms of harmonic functions, admits a noncommutative analogue in the separable case. We also explore the relation between classical and quantum Poisson boundaries by investigating the spectrum of the quantum group. We apply this machinery to find a concrete realization of the Poisson boundaries of the compact quantum group SUq(2) arising from measures on its spectrum.

An uncertainty principle for unimodular quantum groups

We present a generalization of Hirschmans entropic uncertainty principle for locally compact Abelian groups to unimodular locally compact quantum groups. As a corollary, we strengthen a well-known uncertainty principle for compact groups, and generalize the relation to compact quantum groups of Kac type. We also establish the complementarity of finite-dimensional quantum group algebras. In the non-unimodular setting, we obtain an uncertainty relation for arbitrary locally compact groups using the relative entropy with respect to the Haar weight as the measure of uncertainty. We also show that when restricted to q-traces of discrete quantum groups, the relative entropy with respect to the Haar weight reduces to the canonical entropy of the random walk generated by the state.

Realization of quantum group Poisson boundaries as crossed products

For a locally compact quantum group

Compact Operators in Regular LCQ Groups

\mathbb{G}

REPRESENTATION OF LEFT CENTRALIZERS FOR ACTIONS OF LOCALLY COMPACT QUANTUM GROUPS

, consider the convolution action of a quantum probability measure

COMPACT ELEMENTS AND OPERATORS OF QUANTUM GROUPS

\mu

On iterated powers of positive definite functions

on

Boundaries of reduced C * C^{*} -algebras of discrete groups

L_\infty(\mathbb{G})

From Quantum Groups to Groups

. As shown by Junge--Neufang--Ruan, this action has a natural extension to a Markov map on

THE CLASSIFICATION PROBLEM FOR FINITELY GENERATED OPERATOR SYSTEMS AND SPACES

\mathcal{B}(L_2(\mathbb{G}))