Explicity Defined Norms on $J{T}_{\ast }$ Spaces

Abstract

We present a Banach space $X_{\alpha }$, equipped with an explicit norm, whose properties resemble those of the predual $J{T}_{\ast }$ 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 $X_{\alpha }^{\ast }$ does not contain an isomorphic copy of ${\ell }_1$. Finally, the bidual $X_{\alpha }^{\ast \ast }$ is also complementably ${\ell }_2$-saturated, and the quotient $X_{\alpha }^{\ast \ast }/X_{\alpha }$ is isometric to $c_0(2^{\mathbb{N}})$.

37 pages