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# Continuity of Convolution and SIN Groups

Published:2017-02-21
Printed: Dec 2017
• Jan Pachl,
Fields Institute, 222 College Street, Toronto, Ontario M5T 3J1, Canada
• Juris Steprāns,
Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada
 Format: LaTeX MathJax PDF

## Abstract

Let the measure algebra of a topological group $G$ be equipped with the topology of uniform convergence on bounded right uniformly equicontinuous sets of functions. Convolution is separately continuous on the measure algebra, and it is jointly continuous if and only if $G$ has the SIN property. On the larger space $\mathsf{LUC}(G)^\ast$ which includes the measure algebra, convolution is also jointly continuous if and only if the group has the SIN property, but not separately continuous for many non-SIN groups.
 Keywords: topological group, SIN property, measure algebra, convolution
 MSC Classifications: 43A10 - Measure algebras on groups, semigroups, etc. 22A10 - Analysis on general topological groups

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