Certain elements in quotients of measure algebras

Abstract:

Let $G$ be a locally compact group, and $M(G)$ the convolution algebra with unit $\delta$ of all bounded Radon measures on $G$. Let $I$ be a left ideal in $M(G)$ and let $C \cap {I^ \bot }$ be the space of all bounded continuous functions $P$ on $G$ with $\int _GPd\mu = 0$ for all $\mu$ in $I$. Suppose that there exists a function ${P_0}$ in $C \cap {I^ \bot }$ such that $||{P_0}|{|_\infty } > \lim {\sup _{x \to \infty }}|{P_0}(x)|$. Let also $\lambda$ and $\mu$ be two measures in $M(G)$ such that $||\lambda + I|| = ||\mu + I|| = 1$, and $(\lambda + I) \ast \mu \subset \delta + 1$. In this paper we prove under these conditions that there exist a complex number $c$ of modulus one and a point ${x_0}$ in $G$ such that $\int _G Pd\mu = cP({x_0})$ for all functions $P$ in $C \cap {I^ \bot }$ with $||P|{|_\infty } \leqq |\int _G Pd\mu | = 1$. This generalizes a theorem of K. deLeeuw and Y. Katznelson.