Strongly amenable involutive representations of involutive Banach algebras

Let (mathfrak{A }) be a Banach (*) -algebra and let (varphi ) be a nonzero self-adjoint character on (mathfrak{A }) . For a   (*) -representation (pi ) of (mathfrak{A }) on a Hilbert space (mathcal{H }) , we introduce and study strong (varphi ) -amenability of (pi ) in terms of certain states on the von Neumann algebra of bounded operators on (mathcal{H }) . We then give some characterizations of this notion in terms of certain positive functionals on (mathfrak{A }) . We finally investigate some hereditary properties of strong (varphi ) -amenability of Banach algebras.