This question, however looks innocent, looks non-trivial to me. Suppose that $X$ and $Y$ are Banach spaces and let $\alpha$ be any reasonable cross-norm on $X\otimes Y$. *Reasonable means that* $\|x\otimes y \| = \|x\|\|y\|$ for $x\in X, y\in Y$ and $\|f\otimes g\|=\|f\|\|g\|$ for $f\in X^*$ and $g\in Y^*$.

Can we conclude that the operator $I_X \otimes P\colon X\otimes_\alpha Y\to X\otimes_\alpha Y$ is bounded if $P$ is a rank-one projection onto any unit vector $y_0\in Y$ given by $Py = g(y)y_0$ where $g\in Y^*$ is such that $g(y_0)=1$?