Inner amenability of locally compact quantum groups

Inner Amenability Of Locally Compact Quantum Groups

Mohammad Reza Ghanei

• Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran

Abstract

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.

Keywords: Banach module; inner amenability; inner fixed point; locally compact quantum group; mixed identity; strict inner amenability

AMSC: 43A07, 46H25, 46L65, 46L89