Explicity Defined Norms on $JT_*$ Spaces

Abstract

We present a Banach space $X_\alpha$, equipped with an explicit norm, whose properties resemble those of the predual $JT_*$ of the James Tree space. In particular, $X_\alpha$ admits a shrinking dyadic basis with the property that the sums of its elements over any segment are uniformly bounded. The space $X_\alpha$ is complementably $\ell_2$-saturated, and its dual $X_\alpha^*$ does not contain an isomorphic copy of $\ell_1$. Finally, the bidual $X_\alpha^{**}$ is also complementably $\ell_2$-saturated, and the quotient $X_\alpha^{**}/X_\alpha$ is isometric to $c_0(2^{\mathbb{N}})$.