Ergodic properties and harmonic functionals on locally compact quantum groups (Q2874726)

Main objects in this paper are locally compact quantum groups \(G\) and right Banach \(L^{1}(G)\)-modules \(X.\) The author studies some notions like amenability and co-amenability of \(G\) via the notion of ergodic properties of \(X,\) where \(G\) is a locally compact quantum group and \(X\) is a right Banach \(L^{1}(G)\)-module. In this paper, \(N(X)\) denotes the set of all \(\xi\in X\) such that \(Inf\{||\xi\cdot f||: f\in P_{1}(G)\}=0.\) A right Banach \(L^{1}(G)\)-module \(X\) has the ergodic property if \(N(X)\) is a closed subspace \(X.\) The author shows that for a closed topologically left introverted subspace \(X\) of \(L^{\infty}(G)\), the existence of a topological right invariant mean is equivalent with the ergodic property of \(X,\) where \(G\) is a locally compact quantum group.NEWLINENEWLINEAs we know amenability is a fundamental property. We also know which locally compact groups have this property. The author shows that for a locally compact quantum group \(G\) amenability is equivalent with the existence of the ergodic property for every right Banach \(L^{1}(G)\)-module \(X\). The co-amenability of a locally compact quantum group \(G\) is studied through the ergodic property of \(X\). In fact for a co-amenable locally compact quantum group \(G\) and a right Banach \(L^{1}(G)\)-module \(X\), \(LUC(X,G)\) has the ergodic property if and only if \(X\) has the ergodic property.NEWLINENEWLINEThe author also investigates the ergodic property of \(X\) via the Hahn-Banach extension property of \(X\). He uses the fixed point property of some particular maps to characterize the ergodic property of \(X\).