Let $(C(t))\_{t \in R}$ be a cosine function in a unital Banach algebra. Weshow that if $sup\_{t\in R}\Vert C(t)-cos(t)\Vert \textless{} 2$ for somecontinuous scalar bounded cosine function $(c(t))\_{t\in \R},$ then the closedsubalgebra generated by $(C(t))\_{t\in R}$ is isomorphic to $\C^k$ for somepositive 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 $t\in R.$
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机译:令$(C(t))\ _ {t \ in R} $是单位Banach代数中的余弦函数。我们表明如果$ sup \ _ {t \ in R} \ Vert C(t)-cos(t)\ Vert \ textless {} 2 $对于连续连续标量有界余弦函数$(c(t))\ _ {t \在\ R},$中,则对于某个正整数$ k,由$(C(t))\ _ {t \ in R} $生成的闭子代数与$ \ C ^ k $同构。 _ {t \ in \ R} \ Vert C(t)-cos(t)\ Vert \ textless {} {8 \ over 3 \ sqrt 3},$或$ c(t)= I $,则$ C (t)= c(t)$ for $ t \ in R. $
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