An abstract approach to the Crouzeix conjecture

Abstract

Let $A$ be a uniform algebra, $\theta :A\to M_n(\mathbb{C})$ be a continuous homomorphism and $\alpha :A\to A$ be an antilinear contraction such that \[ \|θ(f)+θ(α(f))^*\|\le 2\|f\| \quad(f\in A). \] We show that $\Vert \theta \Vert \le 1+\sqrt{2}$, and that $1+\sqrt{2}$ is sharp. We conjecture that, if further $\alpha (1)=1$, then we may conclude that $\Vert \theta \Vert \le 2$. This would yield a positive solution to the Crouzeix conjecture on numerical ranges. In support of our conjecture, we prove that it is true in two special cases. We also discuss a completely bounded version of our conjecture that brings into play ideas from dilation theory.