Are all uniform algebras AMNM?

A Banach algebra alpha is AMNM if whenever a linear functional phi on alpha and a positive numbers satisfy \phi(ab)-phi(a)phi(b)\ less than or equal to delta parallel to a parallel to . parallel to b parallel to for all a, b is an element of alpha, there is a multiplicative linear functional psi on alpha such that parallel to phi-psi parallel to = o(1) as delta --> 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.