Amenability and weak amenability of second conjugate Banach algebras

Source abstract

For a Banach algebra A \mathfrak {A} , amenability of A ∗ ∗ \mathfrak {A}^{**} necessitates amenability of A \mathfrak {A} , and similarly for weak amenability provided A \mathfrak {A} is a left ideal in A ∗ ∗ \mathfrak {A}^{**} . For G \mathfrak {G} a locally compact group, indeed more generally, L 1 ( G ) ∗ ∗ L^1(\mathfrak {G})^{**} is amenable if and only if G \mathfrak {G} is finite. If L 1 ( G ) ∗ ∗ L^1(\mathfrak {G})^{**} is weakly amenable, then M ( G ) M(\mathfrak {G}) is weakly amenable.