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 $g(y\_0)=1$?



Complementation in tensor products
Tensor products usually make me nervous, but this seems straightforward. Fix a finite sum $\sum x\_j\otimes y\_j$, then
\begin{align\*}
\| I\_X\otimes P(\sum x\_j\otimes y\_j) \|\_\alpha &= \|\sum g(y\_j)x\_j \otimes y\_0\|\_\alpha \\
&= \|\sum g(y\_j)x\_j\|\_X \|y\_0\|\_Y \\
&=\sup\_{f\in X^\*, \|f\|\leq 1} \left|\sum f(x\_j)g(y\_j)\right| \|y\_0\|\\
&= \sup |(f\otimes g)(\sum x\_j\otimes y\_j)| \|y\_0\| \\
&\leq \|g\|\_{X^\*} \|y\_0\| \| \sum x\_j\otimes y\_j\|\_\alpha.
\end{align\*}


