A short proof of the zero-two law for cosine functions

TL;DR

This paper provides a concise proof of the zero-two law for cosine functions in unital Banach algebras, establishing conditions under which the cosine function converges to the identity.

Contribution

It offers a simplified proof of the zero-two law for cosine functions, improving understanding and accessibility of this result in functional analysis.

Findings

If lim sup_{t→0} |C(t)-1_A| < 2, then lim sup_{t→0} ||C(t)-1_A||=0

The proof simplifies previous approaches to the zero-two law

Clarifies the behavior of cosine functions near zero in Banach algebras

Abstract

Let $(C(t))\in\mathbb{R}}$ be a cosine function in a unital Banach algebra. We give a simple proof of the fact that if lim sup$\_{t\to 0}\vert C(t)-1\_A\vert\textless{}2,$ then $lim sup\_{t\to 0}\Vert C(t)-1\_A\Vert=0.$