Weak* fixed point property of reduced Fourier-Stieltjes algebra and generalization of Baggett's theorem

In this paper, we show that if the reduced Fourier-Stieltjes algebra Bρ(G) of a second countable locally compact group G has either weak* fixed point property or asymptotic center property, then G is compact. As a result, we give affirmative answers to open problems raised by Fendler and et al. in 2013. We then conclude that a second countable group with a discrete reduced dual must be compact. This generalizes a theorem of Baggett. We also construct a compact scattered Hausdorff space Ω for which the dual of the scattered C*-algebra C(Ω) lacks weak* fixed point property. The C*-algebra C(Ω) provides a negative answer to a question of Randrianantoanina in 2010. In addition, we prove a variant of Bruck's generalized fixed point theorem for the preduals of von Neumann algebras. Furthermore, we give some examples emphasizing that the conditions in Bruck's generalized conjecture (BGC) can not be weakened any more.