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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Certain elements in quotients of measure algebras

Author: Sadahiro Saeki
Journal: Proc. Amer. Math. Soc. 38 (1973), 437-440
MSC: Primary 43A10
MathSciNet review: 0338689
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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 $ \vert\vert{P_0}\vert{\vert _\infty } > \lim {\sup _{x \to \infty }}\vert{P_0}(x)\vert$. Let also $ \lambda $ and $ \mu $ be two measures in $ M(G)$ such that $ \vert\vert\lambda + I\vert\vert = \vert\vert\mu + I\vert\vert = 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 $ \vert\vert P\vert{\vert _\infty } \leqq \vert\int _G Pd\mu \vert = 1$. This generalizes a theorem of K. deLeeuw and Y. Katznelson.

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Keywords: Locally compact group, measure, ideal
Article copyright: © Copyright 1973 American Mathematical Society