Almost multiplicative functionals. (English) Zbl 0897.46043

A linear functional \(F\) on a Banach algebra is multiplicative if \(F(ab) - F(a)F(b) = 0\) for all \(a\) and \(b\) and almost multiplicative if the above difference is bounded by a universal constant times the product of the norms of \(a\) and \(b\). An algebra \(A\) is called functionally stable (or \(f\)-stable) if any almost multiplicative functional on \(A\) is close to a multiplicative one. (B. E. Johnson used in 1986 the name AMNM algebras).
In the present paper the author proves that any uniform algebra with one generator as well as some Banach algebras of rational or analytic functions are f-stable. It is also shown that the quotient of \(H^{\infty}\) by \(BH^{\infty}\), \(B\) a Blaschke product, is \(f\)-stable if and only if \(B\) is a product of finitely many interpolating Blaschke products.
The paper ends with a list of eight interesting open problems. Sample : Is \(H^{\infty}\) \(f\)-stable?.


46J10 Banach algebras of continuous functions, function algebras
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