Resumen de Uniform continuity over locally compact quantum groups

Volker Runde

• 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 ¡Ä (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 B¢¥edos.Tuset: if G is co-amenable, then the existence of a left invariant mean onM(C0(G)) is sufficient for G to be amenable. Furthermore, we study the space WAP(G) of weakly almost periodic elements of L ¡Ä (G): it is a closed operator system in L ¡Ä (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.