Measure algebras and their second duals
نویسندگان
چکیده
and similarly for the right topological centre Z(r)(A′′). The algebra A is said to be Arens regular if Z(`)(A′′) = Z(r)(A′′) = A′′ and strongly Arens irregular if Z(`)(A′′) = Z(r)(A′′) = A. For example, every C∗-algebra is Arens regular [2]. There has been a great deal of study of these two algebras, especially in the case where A is the group algebra L(G) for a locally compact group G. Results on the second dual algebras of L(G) are given in [2, 16, 17], for example. More recently, the three participants have studied [5] the second dual of a semigroup algebra; here S is a semigroup, and our Banach algebra is A = (` (S), ?). We see that the second dual A′′ can be identified with the space M(βS) of complex-valued, regular Borel measures on βS, the Stone–Cech compactification of S. In fact, (βS, ) is itself a subsemigroup of (M(βS), ). (See [15] for background on (βS, ).) Let A be a Banach algebra which is strongly Arens irregular, and let V be a subset of A′′. Then V is determining for the topological centre if Φ ∈ A for each Φ ∈ A′′ such that Φ Ψ = Φ ♦ Ψ (Ψ ∈ V ). Recently it has become clear that various ‘small’ subsets of A′′ are determining for the topological centre in the case of some of the above algebras. For example, in [5], we showed that, for a wide class of semigroups including all cancellative semigroups, there are just two points in the space βS that are determining for the topological centre of ` 1(S)′′. For an extension of these results to the case of various weighted convolution algebras, see [4] and [3]. Let G be a locally compact group. The measure algebra M(G) of G has also been much studied. This algebra is the multiplier algebra of the group algebra L(G). Even in the case where G is the circle group T,
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