= Solution
A <primitive ideal> is the kernel of an algebraically irreducible unital representation, equivalently the <annihilator of a module> for a nonzero <simple module>. Irreducibility here excludes all proper nonzero invariant linear subspaces, not just closed ones. In a <Banach algebra> the simple modules can be represented as $A/L$ with $L$ a <maximal left ideal>. Such an ideal is closed: if its closure were all of $A$, it would contain an element close enough to $1$ to be invertible by the <Neumann series>, contradicting properness. Maximality then makes its closure itself. Thus the quotient carries a Banach-space norm and a bounded left action.
Let $P$ be a maximal proper two-sided <ideal>. The quotient ring $R=A/P$ is nonzero, unital and simple as a ring. By the <Zorn lemma>, choose a maximal left ideal $L$ of $R$. The <simple module> $R/L$ has an annihilator which is a proper two-sided ideal of $R$. Simplicity forces that annihilator to be zero. Its pullback to $A$ is exactly $P$, proving \b[every maximal proper two-sided ideal is primitive]. If one wants the Banach representation explicitly, realize the same module as the quotient of $A$ by the pullback maximal left ideal, which is closed by the preceding argument.
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