A major trend in noncommutative harmonic analysis is to investigate function spaces related to Fourier analysis (and representation theory) of nonabelian groups. The Fourier algebra, Rajchman algebra and FourierSteiltjes algebra, which are associated with the regular representation, the $C_0$ universal representation and the universal representation of the ambient group respectively, are important examples of such function spaces. These function algebras encode the properties of the group in various ways; for instance the nonexistence of derivations on such algebras indicates their lack of analytic properties, which in turn translates into forms of either commutativity or discreteness for the group itself. In this talk, we study some Banach algebra properties of these function algebras. In particular, we present explicit constructions of continuous derivations on the Fourier algebras of two important matrix groups, namely the group of ${\mathbb R}$affine transformations and the Heisenberg group. Using the structure theory of Lie groups, we extend our results to semisimple Lie groups and nilpotent Lie groups. If time permits, we will discuss weighted versions of the Fourier algebra, called the BeurlingFourier algebra, and some of their properties.

