A Banach algebra a is AMNM if whenever a linear functional φ on a and a positive number δ satisfy |φ(ab)−φ(a)φ(b)|⩽δ‖a‖·‖b‖ for all a, b∈ a, there is a multiplicative linear functional ψ on a such that ‖φ−ψ‖=o(1) as δ→0. K. Jarosz [1] asked whether every Banach algebra, or every uniform algebra, is AMNM. B. E. Johnson [3] studied the AMNM property and constructed a commutative semisimple Banach algebra that is not AMNM. In this note we construct uniform algebras that are not AMNM. 1991 Mathematics Subject Classification 46J10.