# 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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