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We give counterexamples to a conjecture of Bourgain, Casazza, Lindenstrauss and Tzafriri that if X has a unique unconditional basis (up to permutation) then so does $c_0\left(X\right)$. We also give some positive results including a simpler proof that $c_0\left({\ell }_1\right)$ has a unique unconditional basis and a proof that $c_0\left({\ell }_{p_n}^{N_n}\right)$ has a unique unconditional basis when $p_n￬1$, $N_{n+1}\ge 2N_n$ and $(p_n-p_{n+1})log{N}_n$ remains bounded.

The main result of this paper states that if a Banach space X has the property that every bounded operator from an arbitrary subspace of X into an arbitrary Banach space of cotype 2 extends to a bounded operator on X, then every operator from X to an L₁-space factors through a Hilbert space, or equivalently $B({\ell }_{\infty },X*)=\Pi ₂({\ell }_{\infty },X*)$. If in addition X has the Gaussian average property, then it is of type 2. This implies that the same conclusion holds if X has the Gordon-Lewis property (in particular X could be a Banach...