Almost multiplicative functions on commutative Banach algebras

作者：Kulkarni S H^*; Sukumar D

来源：Studia Mathematica, 2010, 197(1): 93-99.

DOI：10.4064/sm197-1-8

• almost multiplicative function
• condition spectrum
• commutative Banach algebras
• carrier space
• Gelfand theory

Let A be a complex commutative Banach algebra with unit 1 and delta > 0. A linear map phi: A -> C is said to be delta-almost multiplicative if
|phi(ab) - phi(a)phi(b)| <= delta parallel to a parallel to parallel to b parallel to for all PH for all a, b is an element of A.
Let 0 < epsilon < 1. The epsilon-condition spectrum of an element a in A is defined by
sigma(epsilon)(a) := {lambda is an element of C: parallel to lambda - a parallel to parallel to(lambda - a)(-1) parallel to >= 1/epsilon}
with the convention that parallel to lambda - a parallel to parallel to(lambda - a)(-1) parallel to when A a is not invertible. We prove the following results connecting these two notions:
(1) If phi(1) = 1 and phi is delta-almost multiplicative, then phi(a) is an element of sigma(delta)(a) for all a in A.
(2) If phi is linear and phi(a) is an element of sigma(epsilon)(a) for all a in A, then phi is delta-almost multiplicative for some delta.
The first result is analogous to the Gelfand theory and the last result is analogous to the classical Gleason-Kahane-Zelazko theorem.

• 出版日期2010