  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.
