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There are two Banach spaces $X,Y$. These spaces have unconditional Schauder bases $\{e_i\}$ and $\{f_i\}$ respectively.

Is this right that $e_i\otimes f_j$ is the unconditional Schauder basis in $X\hat\otimes Y$ (the completion of tensor product which is endowed with projective topology)?

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You find a counterexample on page 90 of the book "Introduction to Tensor Products of Banach Spaces" by Raymond A. Rya.

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