Abstract
M.Sc.
One of the earliest results (1955) in the theory of derivations is the celebrated
theorem of I. M. Singer and J. Wermer [14] which asserts that every bounded
derivation on a commutative Banach algebra has range contained in the
radical. However, they immediately conjectured that their result will still
hold if the boundedness condition was dropped. This conjecture of Singer and
Wermer was confirmed only in 1988, by M. P. Thomas [23], when he showed
that every derivation (bounded or unbounded) on a commutative Banach
algebra has range contained in the radical. But it is not known whether an
analogue of the Kleinecke-Shirokov Theorem holds for everywhere defined
unbounded derivation.