Uniform continuity over locally compact quantum groups
نویسنده
چکیده
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 algebraM(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∞(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.
منابع مشابه
Reiter’s Properties for the Actions of Locally Compact Quantum Goups on von Neumann Algebras
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