Vol. 32, No. 3, 1970

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On the coset ring and strong Ditkin sets

Bertram Manuel Schreiber

Vol. 32 (1970), No. 3, 805–812

We present a complete description of the closed sets in the coset ring (G) of an abelian topological group G. Using this result we show that every such set in a separable, metrizable, locally compact, abelian group Γ is a strong Ditkin set in the sense of Wik, yielding the converse of a theorem of Rosenthal and thus completing the characterization of the strong Ditkin sets with void interior for certain choices of Γ. These two results were first obtained by J. E. Gilbert. Our development of the former rests on the following theorem, which seems to be of independent interest: If φ : G G is a homomorphism and A ∈ℛ(G), then φ(A) ∈ℛ(G).

Mathematical Subject Classification
Primary: 42.56
Secondary: 46.00
Received: 1 April 1969
Published: 1 March 1970
Bertram Manuel Schreiber