The main research in automatic continuity has been on the continuity properties of a homomorphism from one Banach algebra into another. Some of the results on the automatic continuity of homomorphisms between Banach algebras are given in this chapter. We begin the chapter with Johnson's theorem on the uniqueness of the complete norm topology on a Banach space that is an irreducible module over a Banach algebra such that algebra multiplication on the module is continuous [59] (§6). From this the continuity of isomorphisms between semisimple Banach algebras follows easily [59]. In Section 7 we prove a result of Kaplansky [74] on the decomposition of a ring (i. e. additive) isomorphism between two semisimple Banach algebras using automatic continuity methods. Both Sections 6 and 7 depend on Theorem 2. 3. Section 8 contains a brief discussion of the relationship between discontinuous derivations and discontinuous automorphisms, and a proof of the existence of discontinuous derivations from the disc algebra into a Banach module over it [28].

Sections 9 to 12 are concerned with homomorphisms from C*-algebras and, in particular, from the Banach algebra of continuous complex valued functions on a compact Hausdorff space. The main technical result is Theorem 9. 3 (Bade and Curtis [7]) on which Sections 10 and 12 are based. Section 10 is devoted to proving the important theorem of Bade and Curtis [7] on the decomposition of homomorphisms from C(Ω) into a Banach algebra into continuous and highly discontinuous parts.