Homomorphisms of commutative Banach algebras and extensions to multiplier algebras with applications to Fourier algebras
Let $A$ and $B$ be semisimple commutative Banach algebras with bounded approximate identities. We investigate the problem of extending a homomorphism $\varphi : A \to B$ to a homomorphism of the multiplier algebras $M(A)$ and $M(B)$ of $A$ and $B$, respectively. Various sufficient conditions in terms of $B$ (or $B$ and $\varphi $) are given that allow the construction of such extensions. We exhibit a number of classes of Banach algebras to which these criteria apply. In addition, we prove a polar decomposition for homomorphisms from $A$ into $A$ with closed range. Our results are applied to Fourier algebras of locally compact groups.