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