Uniform continuity over locally compact quantum groups

TL;DR

This paper extends the concept of uniform continuity to locally compact quantum groups, exploring the structure of associated operator systems and their implications for amenability and related properties.

Contribution

It introduces a new definition of left uniformly continuous elements for quantum groups, connecting classical and quantum cases, and investigates their algebraic and operator system structures.

Findings

$LUC(G)$ is an operator system containing $C_0(G)$ and contained in $M(C_0(G))$

Partial answer to the open problem on amenability and invariant means for co-amenable quantum groups

Under certain conditions, $LUC(G)$ forms a $C^*$-algebra, especially when $G$ is a group

Abstract

We define, for a locally compact quantum group $G$ in the sense of Kustermans--Vaes, the space of $LUC(G)$ of left uniformly continuous elements in $L^\infty(G)$. This definition covers both the usual left uniformly continuous functions on a locally compact group and Granirer's uniformly continuous functionals on the Fourier algebra. We show that $LUC(G)$ is an operator system containing the $C^*$-algebra $C_0(G)$ and contained in its multiplier algebra $M(C_0(G))$. We use this to partially answer an open problem by Bedos--Tuset: if $G$ is co-amenable, then the existence of a left invariant mean on $M(C_0(G))$ is sufficient for $G$ to be amenable. Furthermore, we study the space $WAP(G)$ of weakly almost periodic elements of $L^\infty(G)$: it is a closed operator system in $L^\infty(G)$ containing $C_0(G)$ and--for co-amenable $G$--contained in $LUC(G)$. Finally, we show that--under certain conditions, which are always satisfied if $G$ is a group--the operator system $LUC(G)$ is a $C^*$-algebra.