Bounded cosine functions close to continuous scalar bounded cosine functions

Abstract

Let $(C(t))\_{t \in R}$ be a cosine function in a unital Banach algebra. We show that if $sup\_{t\in R}\Vert C(t)-cos(t)\Vert \textless{} 2$ for some continuous scalar bounded cosine function $(c(t))\_{t\in \R},$ then the closed subalgebra generated by $(C(t))\_{t\in R}$ is isomorphic to $\C^k$ for some positive integer $k.$ If, further, $sup\_{t\in \R}\Vert C(t)-cos(t)\Vert \textless{} {8\over 3\sqrt 3},$ or if $c(t)=I$, then $C(t)=c(t)$ for $t\in R.$