Title: Modules over subalgebras of the disk algebra

Authors: Vern I. Paulsen and Dinesh Singh

Issue: Volume 55 (2006), Issue 5, 1751-1766

Abstract: This paper deals with the problem of characterizing submodules of $\mathbf{C}(\mathbf{T})$ over certain subalgebras of the disk algebra $\mathbf{A}$. We obtain results that are analogues of the classical characterizations of subspaces of $\mathbf{C}(\mathbf{T})$ that are invariant under multiplication by $z$, i.e., that are submodules over $\mathbf{A}$. These characterizations yield generalizations of Wermer's maximality theorem applicable to these subalgebras. We also present an invariant subspace theorem that seems to be of independent interest and show the equivalence of the classical theorem of the brothers Riesz on analytic measures with the theorem of Fatou.