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2011 | 202 | 3 | 243-259
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On the size of quotients of function spaces on a topological group

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For a non-precompact topological group G, we consider the space C(G) of bounded, continuous, scalar-valued functions on G with the supremum norm, together with the subspace LMC(G) of left multiplicatively continuous functions, the subspace LUC(G) of left norm continuous functions, and the subspace WAP(G) of weakly almost periodic functions. We establish that the quotient space LUC(G)/WAP(G) contains a linear isometric copy of $ℓ_{∞}$, and that the quotient space C(G)/LMC(G) (and a fortiori C(G)/LUC(G)) contains a linear isometric copy of $ℓ_{∞}$ when G is a normal non-P-group. When G is not a P-group but not necessarily normal we prove that the quotient is non-separable. For non-discrete P-groups, the quotient may sometimes be trivial and sometimes non-separable. When G is locally compact, we show that the quotient space LUC(G)/WAP(G) contains a linear isometric copy of $ℓ_{∞}(κ(G))$, where κ(G) is the minimal number of compact sets needed to cover G. This leads to the extreme non-Arens regularity of the group algebra L¹(G) when in addition either κ(G) is greater than or equal to the smallest cardinality of an open base at the identity e of G, or G is metrizable. These results are improvements and generalizations of theorems proved by various authors along the last 35 years and until very recently.
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  • Département de Mathématiques, Université de Rouen, UMR CNRS 6085, Avenue de l'Université, BP.12, F-76801 Saint-Étienne-du-Rouvray, France
  • Department of Mathematical Sciences, University of Oulu, 90014 Oulu, Finland
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