Almost multiplicative functions on commutative Banach algebras
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Date
2010
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Abstract
Let A be a complex commutative Banach algebra with unit 1 and ? > 0. A linear map ?: A ?C is said to be ?-almost multiplicative if |?(ab) - ? (a) ? (b)| ? ?
a
b
for all a, b ? A. Let 0 < e < 1. The e-condition spectrum of an element a in A is defined by ?<inf>e</inf>.(a) := {? ? C:
?-a
?- a
-1
? 1/e} with the convention that
?- a
(? - a)-1
= ? when ? - a is not invertible. We prove the following results connecting these two notions: (1) If ?(1) = 1 and ? is ?-almost multiplicative, then ?(a) ? ?<inf>e</inf>(a) for all a in A.then (2) If ?is lenear and ?(a) ??<inf>e</inf>(a) for all a in A ,then ?-is ? almost multiplicative for some ?. The first result is analogous to the Gelfand theory and the last result is analogous to the classical Gleason-Kahane- ?elazko theorem.
a
b
for all a, b ? A. Let 0 < e < 1. The e-condition spectrum of an element a in A is defined by ?<inf>e</inf>.(a) := {? ? C:
?-a
?- a
-1
? 1/e} with the convention that
?- a
(? - a)-1
= ? when ? - a is not invertible. We prove the following results connecting these two notions: (1) If ?(1) = 1 and ? is ?-almost multiplicative, then ?(a) ? ?<inf>e</inf>(a) for all a in A.then (2) If ?is lenear and ?(a) ??<inf>e</inf>(a) for all a in A ,then ?-is ? almost multiplicative for some ?. The first result is analogous to the Gelfand theory and the last result is analogous to the classical Gleason-Kahane- ?elazko theorem.
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Keywords
Almost multiplicative function, Carrier space, Commutative Banach algebras, Condition spectrum, Gelfand theory
Citation
Studia Mathematica, 2010, 197, 1, pp. 93-99