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Approximately multiplicative functionals. (English) Zbl 0625.46059
Let ${\frak A}$ be a commutative Banach algebra with dual ${\frak A}\sp*$. For $\phi \in A\sp*$, define ${\breve \phi}$(a,b)$=\phi (ab)- \phi (a)\phi (b)$, and call $\phi\delta$-multiplicative iff $\Vert {\breve \phi}\Vert \le \delta$. ${\frak A}$ is an algebra in which approximately multiplicative functionals are near multiplicative (AMNM) if for each $\epsilon >0$, there is $\delta >0$ such that $\inf \{\Vert \phi -\psi \Vert:\psi$ is a $character\}<\epsilon$ whenever $\phi$ in ${\frak A}\sp*$ is $\delta$-multiplicative. The author studies these entities and shows that AMNM algebras include the well-known examples (finite dimensional, $C\sb 0(X)$, $L\sp 1(G)$, $\ell\sp 1({\bbfZ})$, 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

46J05General theory of commutative topological algebras
46J40Structure, classification of commutative topological algebras
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