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Journal of the London Mathematical Society Advance Access originally published online on May 12, 2009
Journal of the London Mathematical Society 2009 80(1):55-71; doi:10.1112/jlms/jdp011
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© 2009 London Mathematical Society

Uniform continuity over locally compact quantum groups

Volker Runde

Department of Mathematical and Statistical Sciences
University of Alberta
Edmonton, Alberta
Canada T6G 2G1
http://www.math.ualberta.ca/~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{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 C0(G) and contained in its multiplier algebra M(C0(G)). We use this to partially answer an open problem by Bédos–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{infty}(G): it is a closed operator system in L{infty}(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.

2000 Mathematics Subject Classification 46L89 (primary); 43A07, 46L07, 46L65, 47L25, 47L50, 81R15 (secondary).

This research supported by NSERC.

Received February 14, 2008; revised September 2, 2008; published online May 12, 2009.


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