Harmonic analysis of Rajchman algebras

Abstract

Abstract harmonic analysis is mainly concerned with the study of locally compact groups, their unitary representations, and the function spaces associated with them. The Fourier and Fourier-Stieltjes algebras are two of the most important function spaces associated with a locally compact group.

The Rajchman algebra associated with a locally compact group is defined to be the set of all elements of the Fourier-Stieltjes algebra which vanish at infinity. This is a closed, complemented ideal in the Fourier-Stieltjes algebra that contains the Fourier algebra. In the Abelian case, the Rajchman algebras can be identified with the algebra of Rajchman measures on the dual group. Such measures have been widely studied in the classical harmonic analysis. In contrast, for non-commutative locally compact groups little is known about these interesting algebras.

In this thesis, we investigate certain Banach algebra properties of Rajchman algebras associated with locally compact groups. In particular, we study various amenability properties of Rajchman algebras, and observe their diverse characteristics for different classes of locally compact groups. We prove that amenability of the Rajchman algebra of a group is equivalent to the group being compact and almost Abelian, a property that is shared by the Fourier-Stieltjes algebra. In contrast, we also present examples of large classes of locally compact groups, such as non-compact Abelian groups and infinite solvable groups, for which Rajchman algebras are not even operator weakly amenable. Moreover, we establish various extension theorems that allow us to generalize the previous result to all non-compact connected SIN-groups.

Finally, we investigate the spectral behavior of Rajchman algebras associated with Abelian locally compact groups, and construct point derivations at certain elements of their spectrum using Varopoulos’ decompositions for Rajchman algebras. Having constructed similar decompositions, we obtain analytic discs around certain idempotent characters of Rajchman algebras. These results, and others that we obtain, illustrate the inherent distinction between the Rajchman algebra and the Fourier algebra of many locally compact groups.