Banach spaces with arbitrary finite Baire order

We investigate intrinsic Baire classes of Banach spaces defined by Argyros, Godefroy and Rosenthal (2003). We introduce a construction, for any Banach space $X$ with a basis, of an $\ell_1$-saturated separable Banach space $Y$ such that for any $\alpha \leqslant \omega_1$ we have $Y^{**}_{1+\alpha} \cong Y \oplus X^{**}_\alpha$, where $X^{**}_\alpha$ denotes the $\alpha$-th intrinsic Baire class of $X$. We apply this construction to answer two open problems by Argyros, Godefroy and Rosenthal (2003), namely we build separable Banach spaces of any Baire order less or equal to $\omega$, and a non-universal separable Banach space of order $\omega_1$. Finally, we apply the construction to show an analogue of a result of Lindenstrauss (1971) by constructing, for any Banach space $X$ with a basis and any $n \in \mathbb{N}$, a Banach space $Y$ such that $Y^{**}_n \cong Y^{**}_{n-1} \oplus X$, showing that any such $X$ can appear as the space of functionals in a bidual Banach space $Y^{**}$ that are of $n$-th intrinsic Baire class but not of $(n-1)$-th intrinsic Baire class.