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Let $X$ denote a specific space of the class of $X_{\alpha ,p}$ Banach sequence spaces which were constructed by Hagler and the first named author as classes of hereditarily ${\ell }_p$ Banach spaces. We show that for $p>1$ the Banach space $X$ contains asymptotically isometric copies of ${\ell }_p$. It is known that any member of the class is a dual space. We show that the predual of $X$ contains isometric copies of ${\ell }_q$ where $\frac{1}{p}+\frac{1}{q}=1$. For $p=1$ it is known that the predual of the Banach space $X$ contains asymptotically isometric copies of $c_0$. Here we...

Hagler and the first named author introduced a class of hereditarily $l_1$ Banach spaces which do not possess the Schur property. Then the first author extended these spaces to a class of hereditarily $l_p$ Banach spaces for $1\le p<\infty$. Here we use these spaces to introduce a new class of hereditarily $l_p\left(c_0\right)$ Banach spaces analogous of the space of Popov. In particular, for $p=1$ the spaces are further examples of hereditarily $l_1$ Banach spaces failing the Schur property.