  We provide a characterization of the Banach spaces $X$ with a Schauder basis
$(e_n)_{n\in\mathbb{N}}$ which have the property that the dual space $X^*$ is
naturally isomorphic to the space $\mathcal{L}_{diag}(X)$ of diagonal operators
with respect to $(e_n)_{n\in\mathbb{N}}$ . We also construct a Hereditarily
Indecomposable Banach space ${\mathfrak X}_D$ with a Schauder basis
$(e_n)_{n\in\mathbb{N}}$ such that ${\mathfrak X}^*_D$ is isometric to
$\mathcal{L}_{diag}({\mathfrak X}_D)$ with these Banach algebras being
Hereditarily Indecomposable. Finally, we show that every $T\in
\mathcal{L}_{diag}({\mathfrak X}_D)$ is of the form $T=\lambda I+K$, where $K$
is a compact operator.
