The classical von Neumann inequality shows that for any contraction T on a Hilbert space, the operator norm of $p(T)$ satisfies
$\|p(T)\| \le \sup_{|z| \le 1} |p(z)|.$
Whereas Ando extended this inequality to pairs of commuting contractions, the corresponding statement for triples of commuting contractions is false.
However, it is still not known whether von Neumann's inequality for triples of commuting contractions holds up to a constant. I will talk about this question and about function theoretic upper bounds for $\|p(T)\|$.