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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 *L*^{1}(*G*)^{**} of its group algebra *L*^{1}(*G*), or the dual LUC(*G*)* where LUC(*G*) is the space of left uniformly continuous functions on *G*.

*Journal Article.*
*0 words.*

*Subjects: *
Pure Mathematics