The Structure Semigroup of Some Convolution Measure Algebras

Second duals of measure algebras

“…A key fact is that Φ M is closed under complex conjugation and this product, so that Φ M is a * -semigroup. This follows from results in [109]; an explicit, simple proof is given in [94]; a result that applies when the group G is replaced by an arbitrary locally compact abelian semigroup with separately continuous product is given in [9] and [96, Theorem (4.1)].…”

“…The structure semigroup of G The structure semigroup of a locally compact abelian group G was introduced by J. L. Taylor in [113] and discussed in some detail by Taylor in [114,Chapters 3,4]; the work is also described in the text [42, §5.1] of Graham and McGehee. This structure semigroup has been used by Brown [6] and by Chow and White [9]; an important early paper of Hewitt and Kakutani is [47].…”

3.—Some Structure Semigroup Results for Arens-Singer Semigroups

“…With this result in mind, we shall identify L t (G + ) with L + for the rest of the paper. The aim of this section is to consider the structure semigroup of L t {G + ) (Taylor 1965) using the approach of Chow and White (1971)-it is clear that L t (G + ) is an L-subalgebra of M(G+) as defined by Chow and White (1971). We point out that the basic idea for the construction originates from F. T. Birtel (Hofmann and Mostert 1966, p. 274) and has also been considered by Ramirez (1968) and Rennison (1969).…”

“…In the third section we show that the structure semigroup (Taylor 1965;Chow and White 1971) of the Lj-algebra of an A-S semigroup is in some sense a compactification of the semigroup and we consider in which cases it is a true compactification. This is a topic mentioned by Taylor (1965) but not developed in detail.…”