convergence of approximate unit

Question

Let A be a Banach algebra and I be a closed ideal.let $\{e_\alpha\}_\alpha$ be an approximate identity for I.Prove that for all $ a \in A $,$\{ae_\alpha\}_\alpha$ is convergence (or example the violation).

Answer 1

Let $A = B(\ell^2)$, the space of bounded operators on $\ell^2$, and let $I = K(\ell^2)$, the space of compact operators. Let $\{P_n\}$ be the usual approximate unit for $I$, ie. $$ P_n((x_j)) = (x_1,x_2,\ldots, x_n,0,0,\ldots) $$ Then, $\{P_n\}$ does not converge in the norm to any operator since, if $n>m$, $$ \|P_n-P_m\| \geq \|P_n(e_n) -P_m(e_n)\| = \|e_n\| = 1 $$ where $e_n$ denotes the standard $n^{th}$ basis vector. Hence, if $T = I$, the identity map, $\{TP_n\}$ does not converge in $A$. In fact, the same argument shows that $\{TP_n\}$ does not converge for any $T \in A\setminus I$.