Let $X$ be a Banach space and $P$ be a projection in $B(X)$. Then $X$ can be renormed so that $P$ has norm $1$.


Can the same be done for a family of projections? That is, given finitely many projections $P\_1$,...,$P\_n$ in $B(X)$ is there an equivalent norm under which all projections become contractive? 


Are there any assumptions other than $P\_i$ commuting that would ensure this?



Renorming a Banach space to make projections contractive
Let $X$ be the Euclidean plane, and consider the two projections
$$
P\_1=\begin{pmatrix}1&A\\0&0\end{pmatrix}\quad\text{and}\quad
P\_2=\begin{pmatrix}0&0\\A&1\end{pmatrix},
$$
where $A$ is a big constant (actually any $A>1$ will do). If a norm on $X$ gave both of them norms $\leq1$, then the same would be true for their product, and for powers of their product,which have the form
$$
(P\_1P\_2)^n=\begin{pmatrix}A^{2n}&A^{2n-1}\\0&0\end{pmatrix}.
$$
But these powers send the vector $\binom10$ to larger and larger vectors, which cannot all be in any ball.


