## 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

### Keywords:

multiplicative functional; AMNM algebra; Lipschitz algebra
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