Almost multiplicative functions on commutative Banach algebras

Abstract

Let A be a complex commutative Banach algebra with unit 1 and δ > 0.We prove the following results connecting these two notions:2010 Mathematics Subject Classification: Primary 46J05.

Approximately spectrum-preserving maps

“…We now emphasise that the only difference of this function with that given in the proof of [27,Theorem 5] is just the factor ϕ(1) −1 . Furthermore, from (4.5) and (4.6) we see that…”

“…for some ε > 0, while in [27] the spectra in (4.1) are replaced by the so-called ε-condition spectra. This section deals with the natural task of replacing the spectra in (4.1) by the pseudospectra in such a way that we generalise both (4.1) and (4.2) by considering the property ϕ(a) ∈ sp ε (a) a ∈ A, a = 1 (4.3)…”

“…for some ε > 0, while in [27] the spectra in condition (I1) are replaced by the so-called ε-condition spectra. On account of those approximate results and the increasing interest in the quantitative aspects of operator theory, it seems natural to ask for approximate versions of Kaplansky's problem.…”

“…We will not consider such operator theoretical refinements either. There are several possible approximate versions of the Gleason-Kahane-Żelazko theorem (see [23,Section 8] and [27]). They are concerned with identifying the approximately multiplicative linear functionals among all linear functionals on a commutative Banach algebra A in terms of spectra.…”

Automatic Continuity of Almost Multiplicative Linear Functionals on Fréchet Algebras

“…Since then, many authors investigated almost multiplicative and nearly (almost) additive maps (operators) between different classes of Banach algebras. See, for example, [1], [8], [11], [15], [16] and [20]. However, as far as we know, the automatic continuity of such maps (operators) between Fréchet algebras has not been studied yet.…”