[Page 1] Elements in the radical of aBanach algebra obeying the unbounded Kleinecke-Shirokov conjecture
نویسندگان
چکیده
It is well known that if D is a bounded derivation on a Banach algebra A and if s is an element of A satisfying [s; Ds] = 0 then Ds must be quasinilpotent. The unbounded Kleinecke-Shirokov conjecture states that the same result holds even if D is unbounded. As yet, the conjecture has neither been proved nor has a case been found where it fails. If there is a counterexample we obtain a further reduction of the problem. We also prove that if s is a quasinilpotent element with essential closed descent one, then [s;Ds] = 0 implies that Ds is quasinipotent. 1991 Mathematics Subject Classi cation: 46J00.
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