Vol. 61, No. 2, 1975

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Continuity of linear maps from Cāˆ—-algebras

Kjeld Laursen

Vol. 61 (1975), No. 2, 483ā€“491

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.

Mathematical Subject Classification 2000
Primary: 46L05
Secondary: 47C10
Received: 30 January 1975
Revised: 4 November 1975
Published: 1 December 1975
Kjeld Laursen