Second duals of measure algebras
نویسنده
چکیده
Let G be a locally compact group. We shall study the Banach algebras which are the group algebra L(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 of 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(Ω). We shall identify the second duals of the measure algebra (M(G), ?) and the group algebra (L(G), ?) as the Banach algebras (M(G̃), 2 ) and (M(Φ), 2 ), respectively, where 2 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, 2 ) 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)′′, 2 ). 2000 Mathematics Subject Classification: Primary 43A10, 43A20; Secondary 46J10
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تاریخ انتشار 2009