Uniform continuity over locally compact quantum groups
by
Volker Runde
University of Alberta

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^∞(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 C₀(G) and contained in its multiplier algebra M(C₀(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(C₀(G)) is sufficient for G to be amenable. Furthermore, we study the space WAP(G) of weakly almost periodic elements of L^∞(G): it is a closed operator system in L^∞(G) containing C₀(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.

Date received: March 21, 2008