The paper proposes a unified approach to many key theorems proved in the last twenty years in different areas of abstract harmonic analysis. This approach is based on the so-called slowly oscillating functions which were introduced in coarse geometry. In addition to this method being the most natural and simple, it also leads to the generalisation of some of the results and to the achievement of some new results. Several of these results concern the topological centres of convolution algebras and semigroup compactifications.
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 ∞, and that the quotient space C(G)/LMC (G) (and a fortiori C(G)/LUC (G)) contains a linear isometric copy of ∞ 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 nondiscrete 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 ∞(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 L 1 (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.
Abstract. We study the Arens regularity of module actions of Banach left or right modules over Banach algebras. We prove that if A has a brai (blai), then the right (left) module action of A on A * is Arens regular if and only if A is reflexive. We find that Arens regularity is implied by the factorization of A * or A * * when A is a left or a right ideal in A * * . The Arens regularity and strong irregularity of A are related to those of the module actions of A on the nth dual A (n) of A. Banach algebras A for which Z(A * * ) = A but A Z t (A * * ) are found (here Z(A * * ) and Z t (A * * ) are the topological centres of A * * with respect to the first and second Arens product, respectively).
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