For a self-adjoint element $a$ in a unital Banach $*$-algebra, is it true that $\|a\| \mathbb{1} -a \geq 0$?

Question

In a Banach $*$-algebra positive elements can be defined as finite sums of the sort $\sum\limits_k b_k^*b_k$, is it true then that, for a self-adjoint $a$, the expression $\|a\|\mathbb{1} -a$ is a positive element in a Banach $*$-algebra, or is it true just for $C^*$-algebras, where arguments can be made about the spectrum to guarantee positivity of $\|a\|\mathbb{1} -a$ ? That is, is there a easy way to write $\|a\|\mathbb{1} -a$ as a finite sum of the form $\sum\limits_k b_k^*b_k$?