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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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