Borel measurable functionals on measure algebras

For Φ,Ψ ∈ A′′, define 〈Φ Ψ, λ〉 = 〈Φ, Ψ · λ〉 (λ ∈ A′) , and similarly for ♦. Thus (A′′, ) and (A′′,♦) are Banach algebras each containingA as a closed subalgebra. The Banach algebra A is Arens regular if and ♦ coincide on A′′, and A is strongly Arens irregular if and ♦ coincide only on A. A subspace X of A′ is left-introverted if Φ · λ ∈ X whenever Φ ∈ A′′ and λ ∈ X . There has been a great deal of study of the two algebras (A′′, ) and (A′′,♦), especially in the case where A is the group algebra (L(G), ? ) or the measure algebra (M(G), ? ) of a locally compact group G. For example, it has been known for a long time that L(G) is strongly Arens irregular for each locally compact group G. On the other hand, each C∗-algebra is Arens regular. 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–Čech compactification of S. In fact, (βS, ) is itself a subsemigroup of (M(βS), ). See [13] for background on (βS, ). Let G be a locally compact group. The algebra M(G) has 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, the Banach algebra M(G) is very complicated; its character space is ‘much larger’ than the dual group Z of T [10]. Starting at a BIRS ‘Research in Teams’ in September, 2006, the three participants have been studying the algebras (M(G)′′, ) and (L1(G)′′, ). Our work continued at other meetings, some at BIRS, and in 2007 and 2008 we established a number of other results that are contained in [6]. The first part of our memoir [6] studied the second dual space of C0(Ω), where Ω is a locally compact space. This second dual is identified with C(Ω̃) for a certain hyper-Stonean space Ω̃; in particular, Ω̃ is compact and extremely disconnected. The space C(Ω̃) contains as a proper closed C∗-subalgebra the space κ(B(Ω)), which is an isometric copy of B(Ω), the space of bounded Borel functions on Ω. The Dixmier