On two refinements of the bounded weak approximate identities

Let $A$ be a commutative Banach algebra with non-empty character space $\mathrm{\Delta }(A)$. In this paper, we change the concepts of convergence and boundedness in the classical notion of bounded approximate identity. This work give us a new kind of approximate identity between bounded approximate identity and bounded weak approximate identity. More precisely, a net $\{e_{\alpha }\}$ in $A$ is a \emph{c-w approximate identity} if for each $a\in A$, the Gel’fand transform of $e_{\alpha }a$ tends to the Gel’fand transform of $a$ in the compact-open topology and we say $\{e_{\alpha }\}$ is \emph{weakly bounded} if the image of $\{e_{\alpha }\}$ under the Gel’fand transform is bounded in $C_0(\mathrm{\Delta }(A))$.