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The celebrated problem
of automatic continuity in Banach algebras—whether or not an arbitrary
homomorphism from the algebra C(X) of all complex continuous functions on a
compact Hausdorff space X is continuous—remains unsolved.
The lack of success on this point has generated quite a bit of effort to determine
‘the extent’ to which a homomorphism is continuous. In the basic work of W. G.
Bade and P. C. Curtis around 1960 it was shown that a homomorphism is necessarily
continuous on some dense subalgebra of the algebra C(X).
Many of these results have later been shown to carry over to a much larger class
of mappings, namely the separable maps (cf. Definition 1.1 below).
Recently, A. M. Sinclair has taken a new look at the homomorphism problems
and succeeded in extending much of Bade’s and Curtis’s work to general C∗-algebras.
In this paper we employ some of Sinclair’s methods and obtain extensions of his
main results, notably we prove (Theorem 3.7) that a separable linear map
defined on a C∗-algebra A is necessarily continuous on a dense subalgebra of
A.
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