Almost-Contractive Maps Between Fourier and Fourier-Stieltjes Algebras

<p><strong>This thesis investigates the Fourier and Fourier-Stieltjes algebras, denoted by A(G) and B(G), associated with a locally compact group G, objects fundamental in the study of abstract harmonic analysis. These algebras are defined as spaces of coefficient functions of representations on locally compact groups. Their Banach dual spaces VN(G) and W*(G) are respectively von Neumann algebras and W*-algebras whose structures depend on the properties of Haar measures and functions of positive type. A norm-gap phenomenon for bounded homomorphisms between Fourier and Fourier-Stieltjes algebras is studied, the phenomenon where such homomorphisms cannot have norms strictly between 1 and 1 + ε. Potential values for ε are explored through multiple techniques. One is a result about almost contractive maps between C*-algebras, namely that they satisfy an almost multiplicative domain principle. Another is a collection of norm-gap estimates for elements of the Banach dual VN(G). A third is the reduction to the case where G is abelian, which is already well-understood in the literature.</strong></p>