Let $A,B$ be two Banach algebra and $A\hat{\otimes} B$ be their projective tensor product. We know that every element $m$ of $A\hat{\otimes} B$ has the following representation: $$m=\sum_{n=1}^\infty x_n\otimes y_n$$ for some $\{x_n\}\subset A,\{y_n\}\subset B$ (which are not unique essentially). Also norm for elements of $A\hat{\otimes} B$ is defined by $$\|m\|=\inf\left\{\sum_{n=1}^\infty \|a_n\|\|b_n\|: m=\sum_{n=1}^\infty a_n\otimes b_n,\qquad \{a_n\}\subset A,\{b_n\}\subset B\right\}.$$ Define $$am=\sum_{n=1}^\infty (ax_n)\otimes y_n,\qquad a\in A.$$ By definition we know $$\|am\|=\inf\left\{\sum_{n=1}^\infty \|a_n\|\|b_n\|: am=\sum_{n=1}^\infty a_n\otimes b_n,\qquad \{a_n\}\subset A,\{b_n\}\subset B\right\}.$$ Now could we conclude the following statement? $$\|am\|=\inf\left\{\sum_{n=1}^\infty \|aa_n\|\|b_n\|: m=\sum_{n=1}^\infty a_n\otimes b_n,\qquad \{a_n\}\subset A,\{b_n\}\subset B\right\}.$$ If we couldn't, under which condition it is possible?

For definition of projective tensor product see the following from "Complete normed algebras" of Bonsall and Duncan.---**Definition 9 to Proposition 12**