## Are all uniform algebras AMNM?(English)Zbl 0865.46035

A Banach algebra $${\mathcal A}$$ is said to be AMNM (approximately multiplicative functionals are near multiplicative functionals) if for every $$\varepsilon >0$$ there is a $$\delta >0$$ such that whenever a linear functional $$\varphi$$ on $${\mathcal A}$$ is $$\delta$$-multiplicative in the sense that $$|\varphi (ab)- \varphi (a) \varphi (b) |\leq \delta|a|\cdot |b|$$ for all $$a$$ and $$b$$ in $${\mathcal A}$$, it follows that there is a multiplicative linear functional $$\psi$$ on $${\mathcal A}$$ such that $$|\varphi- \psi|<\varepsilon$$. K. Jarosz [Perturbations of Banach algebras, Lect. Notes Math. 1120, Springer (1985; Zbl 0557.46029)] asked whether every (commutative) Banach algebra, or every uniform algebra, is AMNM. B. E. Johnson [J. Lond. Math. Soc., II. Ser. 34, 489-510 (1986; Zbl 0625.46059)] 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.

### MSC:

 46J10 Banach algebras of continuous functions, function algebras

### Citations:

Zbl 0557.46029; Zbl 0625.46059
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