On the isomorphism of the spaces $\mathcal{k}$ and $\mathcal{K}$

Abstract

Let $1 \leqslant p \leqslant q < \infty$, let $\mathsf{X}$ be a separable $\mathcal{L}_p$-space and let $\mathsf{Y}$ be a separable $\mathcal{L}_q$-space. We show that the space $\mathfrak{L}(\mathsf{X},\mathsf{Y})$ of bounded linear operators from $\mathsf{X}$ into $\mathsf{Y}$ is isomorphic to $\mathfrak{L}(\ell_p,\ell_q)$ if, and only if, $\mathsf{Y}$ is complemented in its bidual. We also show that $\mathfrak{L}(\mathsf{Z},\ell_{\infty})$ is isomorphic to $\mathfrak{L}(c_0,\ell_{\infty})$ for every separable $\mathcal{L}_{\infty}$-space $\mathsf{Z}$. This extends and complements the earlier result of Arias and Farmer [On the structure of tensor products of $\ell_p$-spaces, Pacific J. Math.]. Based on this general result, we give a completely different proof of the main result of Astashkin, Leśnik and Maligranda [Isomorphic structure of Cesàro and Tandori spaces, Canad. J. Math.], which states that the space $\mathcal{K}$ and its sequence counterpart $\mathcal{k}$ are isomorphic. Furthermore, we complete the picture by showing that the spaces $\mathcal{k}$ and $\mathcal{K}$, although isomorphic, are not isometric. Along the way, we show that for every infinite compact metrizable space $K$ the $\ell_{\infty}$-sum of copies of the space $\mathscr{M}(K)$ of regular Borel measures on $K$ is isomorphic to $( \bigoplus_{n=1}^{\infty} L_1[0,1])_{\ell_{\infty}} \approx \mathfrak{L}(\ell_1,L_1[0,1])$. This provides an abstract blueprint for another result of Astashkin, Leśnik and Maligranda, namely that $\mathcal{K}$ is isomorphic to $( \bigoplus_{n=1}^{\infty} \mathscr{M}[0,1])_{\ell_{\infty}}$. Finally, we show that there are models of set theory without the Axiom of Choice in which the spaces $\mathcal{k}$ and $\mathcal{K}$ are no longer isomorphic. This last part was inspired by Väth's paper [The dual space of $L_{\infty}$ is $L_1$, Indag. Math. (N.S.)].