| dc.description.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. | en |
