INNER AMENABILITY OF LOCALLY COMPACT QUANTUM GROUPS
We initiate a study of inner amenability for a locally compact quantum group 𝔾 in the sense of Kustermans–Vaes. We show that all amenable and co-amenable locally compact quantum groups are inner amenable. We then show that inner amenability of 𝔾 is equivalent to the existence of certain functionals associated to characters on L1(𝔾). For co-amenable locally compact quantum groups, we introduce and study strict inner amenability and its relation to the extension of the co-unit ϵ from C0(𝔾) to L∞(𝔾). We then obtain a number of equivalent statements describing strict inner amenability of 𝔾 and the existence of certain means on subspaces of L∞(𝔾) such as LUC(𝔾), RUC(𝔾) and UC(𝔾). Finally, we offer a characterization of strict inner amenability in terms of a fixed point property for L1(𝔾)-modules.