Hereditarily indecomposable Banach algebras of diagonal operators

We provide a characterization of the Banach spaces X with a Schauder basis (e(n))(n is an element of N) which have the property that the dual space X* is naturally isomorphic to the space L-diag(X) of diagonal operators with respect to (e(n))(n is an element of N). We also construct a Hereditarily Indecomposable Banach space X-D with a Schauder basis (e(n))(n is an element of N) such that X-D* is isometric to L-diag(X-D) with these Banach algebras being Hereditarily Indecomposable. Finally, we show that every T is an element of L-diag(X-D) is of the form T = lambda I + K, where K is a compact operator.