Almost multiplicative functions on commutative Banach algebras
Volume 197 / 2010
Abstract
Let $A$ be a complex commutative Banach algebra with unit 1 and $\delta >0$. A linear map $\phi:A\rightarrow\mathbb C$ is said to be $\delta$-almost multiplicative if $$|{\phi(ab)-\phi(a)\phi(b)}|\leq \delta \|{a}\|\,\|{b}\| \quad \hbox{for all} \ a,b \in A.$$ Let $0<\epsilon<1$. The $\epsilon$-condition spectrum of an element $a$ in $A$ is defined by $$\sigma_\epsilon(a):=\{\lambda\in \mathbb C : \|{\lambda-a}\|\,\|{(\lambda-a)^{-1}}\| \geq {1}/{\epsilon}\}$$ with the convention that $\|{\lambda-a}\|\,\|{(\lambda-a)^{-1}}\|=\infty$ when $\lambda-a$ is not invertible. We prove the following results connecting these two notions:
(1) If $\phi(1) = 1$ and $\phi$ is $\delta$-almost multiplicative, then $\phi(a)\in\sigma_{\delta}(a)$ for all $a$ in $A$.
(2) If $\phi$ is linear and $\phi(a)\in\sigma_{\epsilon}(a)$ 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–Żelazko theorem.
