Document Type : Research Paper

Author

• Bahman Hayati

Faculty of Mathematical Sciences, Malayer University, P. O. Box 16846-13114, Malayer, Iran

Abstract

For a Banach algebra $\mathfrak{A}$, we introduce ~$c.c(\mathfrak{A})$, the set of all $\varphi \in {\mathfrak{A}}^{\ast }$ such that ${\theta }_{\varphi }:\mathfrak{A}\to {\mathfrak{A}}^{\ast }$ is a completely continuous operator, where ${\theta }_{\varphi }$ is defined by ${\theta }_{\varphi }(a)=a\cdot \varphi$ for all $a\in \mathfrak{A}$. We call $\mathfrak{A}$, a completely continuous Banach algebra if $c.c(\mathfrak{A})={\mathfrak{A}}^{\ast }$. We give some examples of completely continuous Banach algebras and a sufficient condition for an open problem raised for the first time by J.E Gale, T.J. Ransford and M. C. White: Is there exist an infinite-dimensional amenable Banach algebra whose underlying Banach space is reflexive? We prove that a reflexive, amenable, completely continuous Banach algebra with the approximation property is trivial.

Keywords

• Amenability
• Completely continuous
• Banach algebra