Extension of Derivations, and Connes- Amenability of the Enveloping Dual Banach Algebra - Choi, Yemon; Samei, Ebrahim; Stokke, Ross (Mathematica Scandinavica, 2015)
If D:A → X is a derivation from a Banach algebra to a contractive, Banach A-bimodule, then one can equip X^** with an A^**-bimodule structure, such that the second transpose D^**: A^** → X^** is again a derivation. We prove an analogous extension result, where A^** is replaced by F(A), the enveloping dual Banach algebra of A, and X^** by an appropriate kind of universal, enveloping, normal dual bimodule of X. Using this, we obtain some new characterizations of Connes-amenability of F(A). In particular we show that F(A) is Connes-amenable if and only if A admits a so-called WAP-virtual diagonal. We show that when A=L^1(G), existence of a WAP-virtual diagonal is equivalent to the existence of a virtual diagonal in the usual sense. Our approach does not involve invariant means for G.