On the Azimi-Hagler Banach spaces (Q1098051)

\textit{P. Azimi} and \textit{J. N. Hagler} [Pac. J. Math. 122, 287--297 (1986; Zbl 0609.46012)] used some ``mixture'' of \(\ell\) 1-norm and the norm of the Lorentz sequence space d(a,1) to construct for every real sequence \((a_ i)^{\infty}_{i=1}\) satisfying the conditions \(1=a_ 1\geq a_ 2\geq...\to 0\) and \(\sum^{\infty}_{i=1}a_ i=\infty\) the Banach space \(X_ a\) with the following properties: (a) \(X_ a\) has boundedly complete basis; (b) \(X_ a\) is hereditarily \(\ell\) 1; (c) \(X_ a\) has codimension 1 in its weak sequential closure in \(X^{**}.\) The author made further steps in the investigation of the spaces \(X_ a:\) 1. \(X_ a\) and \(X_ b\) are isomorphic if and only if they are equal as sets. 2. Every complemented subspace of \(X_ a\) which is not weakly sequentially complete contains a complemented subspace which is isomorphic to \(X_ a\).