Kernels of surjections from ${\cal L}_1$-spaces with an application to Sidon sets

Abstract

If $Q$ is a surjection from $L^1(\mu)$, $\mu$ $\sigma$-finite, onto a Banach space containing $c_0$ then (*) $\ker Q$ is uncomplemented in its second dual. If $Q$ is a surjection from an ${\cal L}_1$-space onto a Banach space containing uniformly $\ell_n^\infty$ ($n=1,2,\dots$) then (**) there exists a bounded linear operator from $\ker Q$ into a Hilbert space which is not 2-absolutely summing. Let $S$ be an infinite Sidon set in the dual group $\Gamma$ of a compact abelian group $G$. Then $L^1_{\tilde{S}}(G)=\{f\in L^1(G): \hat{f}(\gamma)=0$ for $\gamma\in S\}$ satisfies (*) and (**) hence $L^1_{\tilde{S}}(G)$ is not an ${\cal L}_1$-space and is not isomorphic to a Banach lattice.