Second duals of measure algebras

H. G. Dales ; A. T.-M. Lau ; D. Strauss

GDML_Books, (2011), p.

Résumé

Let G be a locally compact group. We shall study the Banach algebras which are the group algebra L¹(G) and the measure algebra M(G) on G, concentrating on their second dual algebras. As a preliminary we shall study the second dual C₀(Ω)” of the C*-algebra C₀(Ω) for a locally compact space Ω, recognizing this space as C(Ω̃), where Ω̃ is the hyper-Stonean envelope of Ω. We shall study the C*-algebra $B^{b}(Ω)$ of bounded Borel functions on Ω, and we shall determine the exact cardinality of a variety of subsets of Ω̃ that are associated with $B^{b}(Ω)$. We shall identify the second duals of the measure algebra (M(G),∗) and the group algebra (L¹(G),∗) as the Banach algebras (M(G̃),□ ) and (M(Φ),□ ), respectively, where □ denotes the first Arens product and G̃ and Φ are certain compact spaces, and we shall then describe many of the properties of these two algebras. In particular, we shall show that the hyper-Stonean envelope G̃ determines the locally compact group G. We shall also show that (G̃,□ ) is a semigroup if and only if G is discrete, and we shall discuss in considerable detail the product of point masses in M(G̃). Some important special cases will be considered. We shall show that the spectrum of the C*-algebra $L^{∞}(G)$ is determining for the left topological centre of L¹(G)”, and we shall discuss the topological centre of the algebra (M(G)”,□ ).

Détail

Zbl 1248.43002

EUDML-ID : urn:eudml:doc:286050