## Approximately multiplicative functionals.(English)Zbl 0625.46059

Let $${\mathfrak A}$$ be a commutative Banach algebra with dual $${\mathfrak A}^*$$. For $$\phi \in A^*$$, define $${\breve \phi}$$(a,b)$$=\phi (ab)- \phi (a)\phi (b)$$, and call $$\phi\delta$$-multiplicative iff $$\| {\breve \phi}\| \leq \delta$$. $${\mathfrak A}$$ is an algebra in which approximately multiplicative functionals are near multiplicative (AMNM) if for each $$\epsilon >0$$, there is $$\delta >0$$ such that $$\inf \{\| \phi -\psi \|:\psi$$ is a $$character\}<\epsilon$$ whenever $$\phi$$ in $${\mathfrak A}^*$$ is $$\delta$$-multiplicative. The author studies these entities and shows that AMNM algebras include the well-known examples (finite dimensional, $$C_ 0(X)$$, $$L^ 1(G)$$, $$\ell^ 1({\mathbb{Z}})$$, disc algebra), but not all. A result of Gleason about multiplicativeness of functions with range contained in the spectrum is studied in a more general context.
Reviewer: E.J.Barbeau

### MSC:

 46J05 General theory of commutative topological algebras 46J40 Structure and classification of commutative topological algebras
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