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An abstract approach to the Crouzeix conjecture

Let $A$ be a uniform algebra, $θ:A\to M_n(\mathbb{C})$ be a continuous homomorphism and $α:A\to A$ be an antilinear contraction such that \[ \|θ(f)+θ(α(f))^*\|\le 2\|f\| \quad(f\in A). \] We show that $\|θ\|\le 1+\sqrt{2}$, and that $1+\sqrt2$ is sharp. We conjecture that, if further $α(1)=1$, then we may conclude that $\|θ\|\le2$. 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.