Bounded Cosine Functions Close to Continuous Scalar Bounded Cosine Functions

摘要

Let (C(t))(t is an element of R) be a cosine function in a unital Banach algebra. We show that if sup(t is an element of R) parallel to C(t) - c(t)parallel to < 2 for some continuous scalar bounded cosine function (c(t))(t is an element of R) then the closed subalgebra generated by (C(t))(t is an element of R) is isomorphic to C-k for some positive integer k. If, further, sup(t is an element of R) parallel to C(t) - c(t)parallel to < 8/3 root 3, then C(t) = c(t) for t is an element of R.