Superreflexivity and property (T) for group actions on Banach spaces

Specialeforsvar ved Anna Munk Ebbesen 

Titel: Superreflexivity and property (T) for group actionson Banach spaces 

Abstract: This master thesis in mathematics is concerned with the study of the analogue property (T_B) of Kazhdan's property (T) for a discrete group Γ when the group acts by invertible isometries on a Banach space B. When B is superreflexive, the closed subspace of Γ-invariant vectors is complemented in B, and techniques similar to those used in the Hilbert space case can be employed. We give a thorough introduction to superreflexivity and study its connections to uniform convexity and uniform smoothness. It is further shown that L_p(μ)-spaces (where μ is a σ-finite measure) are superreflexive, and that they posess a rich group of linear isometries. Using the Mazur map between L_p(μ)-spaces, we show that property (T) implies property (T_Lp(μ)).

 Vejleder:   Magdalene Musat
Censor:     Søren Møller, SDU