Bounded cosine functions close to continuous scalar bounded cosine functions

TL;DR

This paper investigates the structure of cosine functions in Banach algebras that are close to the scalar cosine function, establishing conditions under which they are algebraically similar or identical.

Contribution

It provides new bounds on how close a Banach algebra cosine function must be to the scalar cosine to determine its algebraic structure or equality.

Findings

If the difference is less than 2, the generated algebra is isomorphic to a finite product of complex numbers.

If the difference is less than 8/(3√3), the cosine function coincides with the scalar cosine.

The results specify conditions for the cosine function to be exactly scalar or structurally similar.

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 $t\in R.$