Ideal generated by a set of singular elements in a Banach algebra.

143 Views Asked by At

Let $A$ be a commutative unital Banach Algebra. Suppose for every $a\in A$, $\|a\|=1$, I get a singular element $b_a$. I know that each such $b_a$ is contained in a proper maximal ideal of $A$. Is it possible that all the $b_a$'s together are in aproper maximal ideal?