Amenability and Modules For Arens Product Algebras

stokke, ross;

Given a Banach algebra A, we introduce the notion of a left dual Banach algebra (LDBA) over A, and we establish that every LDBA over A is a left Arens product algebra over A. This can be viewed as a Banach algebraic version of the fact that every semigroup compactification is a Gelfand compactification. We show how A-module operations can be extended to obtain module operations for left Arens product algebras over A that satisfy attractive w ∗-continuity properties. We introduce a notion of left Connes amenability for LDBAs, and show that the amenability of a locally compact group G is equivalent to left Connes amenability of either the bidual L1(G) ∗∗ of its group algebra L1(G), or the dual LUC(G) ∗ where LUC(G) is the space of left uniformly
continuous functions on G.