Abstract
Abstract The theory of automatic continuity deals with the extent to which certain algebraic conditions on linear mappings imply their continuity. Typical results in this theory establish the continuity of all multiplicative linear functionals on Banach algebras and of all positive linear operators on Banach lattices. The theory is particularly well developed in the context of Banach algebras. In fact, the automatic continuity problem for homomorphisms and derivations on certain classes of Banach algebras is intimately related to the structure of general and particular Banach algebras, and has led, over the last four decades, to remarkable new insight into the theory of Banach algebras. The state of the art is well documented in the comprehensive account of Dales (2000).
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