Almost multiplicative functions on commutative Banach algebras

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 ?e.(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) ? ?e(a) for all a in A.then (2) If ?is lenear and ?(a) ??e(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.