An Introduction to Fourier Algebras of Locally Compact Groups

Mahya Ghandehari
University of Delaware

Sunday, July 16, 2017,   11:00 - 12:30
Sunday, July 16, 2017,   14:00 - 15:30

VENUE   Lecture Hall 2, Niavaran Bldg.


A major trend in non-commutative harmonic analysis is to investigate function spaces related to Fourier analysis (and representation theory) of non-abelian groups. The Fourier algebra and the Fourier-Steiltjes algebra, which are associated with the regular 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 non-existence of derivations on such algebras indicates their lack of analytic properties. In these lectures, we overview some important properties of these function algebras. For a locally compact group $G$, the Fourier-Stieltjes algebra of $G$, denoted by $B(G)$, is the set of all the matrix coefficient functions of $G$ equipped with pointwise algebra operations. Eymard proved that $B(G)$ can be identified with the dual of the group $C^*$-algebra of $G$. Moreover, the Fourier-Stieltjes algebra together with the norm from the above duality turns out to be a Banach algebra. We also study subspaces of $B(G)$, called $A_pi(G)$, generated by all the matrix coefficient functions of $G$ associated with a fixed unitary representation $\pi$. As an important example of such subspaces, we study the Rajchman algebra of $G$, denoted by $B_0(G)$, which is the set of elements of the Fourier-Stieltjes algebra that vanish at infinity.