Strongly amenable involutive representations of involutive Banach algebras

Abstract

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.