2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/56966We estimate the norm of the almost Mathieu operator $H_{θ,λ} =U+U^*+(λ/2)(V+V^*)$ in the rotation $C^*$-algebra $A_θ=C^*(U,V unitaries;UV=e^{2πiθ} VU)$. In this process, we significantly improve the inequality $\| H_θ\| \leq 2\sqrt{2}$, $θ\in [0.25,0.5]$, conjectured by Beguin, Valette and Zuk.18 pages,3 figuresMathematical PhysicsOperator AlgebrasSpectral Theory47B36;47A30;81Q10;46L60text