In these notes we are concerned with algebraic conditions on a linear operator from one Banach space into another that force the continuity of the linear operator. The main results are in the theory of Banach algebras, where the continuity of homomorphisms under suitable hypotheses is part of the standard theory (see Rickart [103], and Bonsall and Duncan [18]). The continuity of a multiplicative linear functional on a unital Banach algebra is the seed from which these results on the automatic continuity of homomorphisms grew, and is typical of the conditions on a linear operator that imply its continuity. Homomorphisms, derivations, and linear operators intertwining with a pair of continuous linear operators are the most important general classes of linear operators whose automatic continuity has been studied. These notes are an attempt to collect together and unify some of the results on the automatic continuity of homomorphisms and intertwining operators.

The most important results in these notes are in sections 4, 6, 8, 9, 10, and 12 of Chapters 2 and 3. The guiding problem behind Chapter 2 is to find necessary and sufficient conditions on a pair (T, R) of continuous linear operators on Banach spaces X, Y, respectively, so that each linear operator S from X into Y satisfying ST = RS is continuous (Johnson [58]). The equivalent problem for homomorphisms is to find necessary and sufficient conditions on a pair of Banach algebras A and B so that each homomorphism from A into B (or onto B) is continuous (Rickart [103, §5]) (Chapter 3).