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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Continuity of translation in the dual of $ L\sp \infty(G)$ and related spaces


Authors: Colin C. Graham, Anthony T. M. Lau and Michael Leinert
Journal: Trans. Amer. Math. Soc. 328 (1991), 589-618
MSC: Primary 43A15; Secondary 43A10, 46L10
MathSciNet review: 1014247
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Abstract: Let $ X$ be a Banach space and $ G$ a locally compact Hausdorff group that acts as a group of isometric linear operators on $ X$. The operation of $ x \in G$ on $ X$ will be denoted by $ {L_x}$. We study the set $ {X_c}$ of elements $ \mu \in X$ such that $ x \mapsto {L_x}\mu $ is continuous with respect to the topology on $ G$ and the norm-topology on $ X$. The spaces $ X$ studied include $ M{(G)^{\ast} },{\text{LUC}}{(G)^{\ast} },{L^\infty }{(G)^{\ast} },{\text{VN}}(G)$, and $ {\text{VN}}{(G)^{\ast} }$. In most cases, characterizations of $ {X_c}$ do not appear to be possible, and we give constructions that illustrate this. We relate properties of $ {X_c}$ to properties of $ G$. For example, if $ {X_c}$ is sufficiently small, then $ G$ is compact, or even finite, depending on the case. We give related results and open problems.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1991-1014247-6
PII: S 0002-9947(1991)1014247-6
Keywords: Amenable groups, continuity of translation, left uniformly continuous functions, measure algebra of a locally compact group, nonmeasurable subgroups of a locally compact group, second dual space of the group algebra, translation-invariant means on locally compact groups, von Neumann algebra
Article copyright: © Copyright 1991 American Mathematical Society