Bounded cosine functions close to continuous scalar bounded cosine functions
Abstract
Let
(C(t))_t∈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∈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∈R.