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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
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