Geometry and fixed point properties for a class of Banach algebras associated to locally compact groups

In this thesis, we discuss two separate topics from the theory of abstract harmonic analysis. The first topic revolves around a locally compact group (Part I), the second one deals with the more abstract setting of a Lau algebra (Part II).;Part I is primarily concerned with the study of Dunford-Pettis operators related to the group algebra L1( G) and the Fourier algebra A(G) of a locally compact group. We provide simpler proofs of known results relating to L1(G) when G is first countable. We then investigate the corresponding properties for A(G) when G is non-abelian. A new Banach space DP(G) associated to G is introduced and we investigate some of its properties. We compare DP(G) to important subspaces of A( G)*. In addition, it is shown that DP(G)* has a natural multiplication, turning it into a Banach algebra. Stronger properties are developed for discrete groups, where weak convergence implies multiplier convergence for sequences. Afterwards, we investigate the tensor product of two abstract Segal algebras and subsequently introduce the concept of a vector-valued Segal algebra.;Part II is of a more abstract nature and relies heavily on the powerful properties of von Neumann algebras. We prove several fixed point theorems characterizing the left amenability of a Lau algebra. We also prove several hereditary properties for left amenable Lau algebras, with applications to semigroups.;The notion of operator left amenability for a Lau algebras is discussed and it is shown to be equivalent to left amenability. To finish, we introduce several new notions of amenability for a Lau algebra.