Journal of Operator Theory

Volume 90, Issue 1, Summer 2023 pp. 209-221.

An abstract approach to the Crouzeix conjecture

Authors: Raphael Clouatre, Maleva Ostermann (2), and Thomas Ransford (3)
Author institution: (1) Department of Mathematics, University of Manitoba, Winnipeg (Manitoba), R3T 2N2, Canada
(2) Departement de mathematiques et de statistique, Universite Laval, Quebec City (Quebec), G1V 0A6, Canada
(3) Departement de mathematiques et de statistique, Universite Laval, Quebec City (Quebec), G1V 0A6, Canada

Summary: 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 \[ \|\theta(f)+\theta(\alpha(f))^*\|\leqslant 2\|f\| \quad(f\in A). \] We show that $\|\theta\|\leqslant 1+\sqrt{2}$, and that $1+\sqrt2$ is sharp. We conjecture that, if further $\alpha(1)=1$, then we may conclude that $\|\theta\|\leqslant 2$. 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.

DOI: http://dx.doi.org/10.7900/jot.2021nov15.2364
Keywords: Crouzeix conjecture, uniform algebra, homomorphism, completely bounded map.

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