Amenability properties of semitopological semigroups and related Banach algebras

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My team and I propose to investigate amenability properties of semi-topological semigroups, centering on how they determine the common fixed point property of the semigroup and how they decide the structure of a related Banach algebra. This is a constitutive component of our long-term objective. For the latter, we aim to integrate ideas/methods in developing the topological group/semigroup theory and the Banach algebra theory. We engage in revealing deep links between the two areas. Amenability theory for groups may trace back to the 1920s when J. von Neumann, studying the Banach-Tarski paradox, raised the question of whether there is an invariant measure for a group acting on certain sets. M.M. Day laid down the foundation of the theory in the 1950s. During the past 70 years, the investigation has been interacting fruitfully with harmonic analysis and Banach algebra theory, giving rise to many beautiful and profound results in the area. The amenability property of a semi-topological semigroup S is expounded in terms of the existence of a left-invariant mean on a subspace of continuous functions on S. When S acts on a topological space, its amenability property determines many features of the action. There are various types of semigroup actions. Among them, affine actions and non-expansive actions are of extreme importance to many analysis areas. These types of actions of S on a weakly compact or weak* compact set of a Banach space are of particular interest, with several long-standing open problems in the background. We will focus on the common fixed point properties of S for such actions, investigating intrinsic relations linking these to the amenability properties of S. After a pioneering work of B.E. Johnson in the 1970s, amenability theory has been extended to the area of Banach algebras, providing inspiring ideas to probe the structure of a Banach algebra. On the other hand, amenability is a very restrictive condition for a Banach algebra. In recent years, generalized/weak versions of amenability for Banach algebras have been introduced in the literature. Regarding them, many crucial problems are pending an intensive investigation. We will concentrate on the problems concerning Banach algebras related to groups and semigroups in this proposal. In particular, we will investigate the weighted group and measure algebras and their center algebras, and weighted measure algebras of semigroups. We will further investigate the class of F-algebras, which includes many important Banach algebras arising from the harmonic analysis. The proposed research will contribute to developing the theory of Banach algebras, harmonic analysis, and general functional analysis. The outcome will have foreseeable applications in dynamic systems, ergodic theory, and approximation theory. Participating in the research, trainees will gain fundamental knowledge and skills in functional analysis that will greatly benefit their future research-related careers.