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Approximately multiplicative functionals on algebras of smooth functions. (English) Zbl 1064.46032

Let \(A^{\ast }\) be the dual of a commutative Banach algebra \(A.\) For each \(\varphi \in A^{\ast }\), define a bilinear function \(B_{\varphi }\) by \( B_{\varphi }( a,b) :=\varphi ( ab) -\varphi ( a) \varphi ( b) \) for all \(a,b\in A.\) Then a Banach algebra is an AMNM algebra if for each \(\varepsilon >0\) there exists \(\delta >0\) such that for each \(\varphi \in A^{\ast }\) with \(\| B_{\varphi }\| <\delta \) there exists a multiplicative functional \(\psi :A\rightarrow \mathbb{C}\) such that \(\| \varphi -\psi \| <\delta . \) Here the abbreviation AMNM stands for: Approximately Multiplicative functionals are Near Multiplicative functionals. The main result of the paper shows that the Banach algebra \(C^{N}[0,1] ^{M}\) of all complex-valued functions defined on \([0,1] ^{M}\) with \(N\)th order continuous partial derivatives is an AMNM algebra. Moreover, this result is extended to certain Lipschitz algebras.

MSC:

46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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