Vol. 33, No. 2, 1970

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ISSN: 0030-8730
On extending almost periodic functions

John Findley Berglund

Vol. 33 (1970), No. 2, 281–289
Abstract

Deleeuw and Glicksberg have shown that every weakly almost periodic function on a subgroup (respectively, a dense subgroup) of a locally compact abelian topological group (respectively, an abelian topological group) extends to the whole group as a weakly almost periodic function. By way of a theorem about extending strongly almost periodic functions on subsemigroups of semitopological semigroups, we show that every almost periodic function on a subgroup (respectively, a dense subgroup) of a locally compact abelian topological group (respectively a topological group) extends to the whole group as an almost periodic function. Furthermore, for the algebra of weakly almost periodic functions on a dense subsemigroup of a semitopological semigroup to consist of restrictions of weakly almost periodic functions on the larger semigroup, we need only require that each weakly almost periodic function on the subsemigroup extend continuously. Analogous statements hold for almost periodic and strongly almost periodic functions on dense subsemigroups of semitopological semigroups.

Mathematical Subject Classification 2000
Primary: 43A60
Secondary: 22A20
Milestones
Received: 9 September 1969
Published: 1 May 1970
Authors
John Findley Berglund