Vol. 61, No. 2, 1975

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Continuity of linear maps from Cāˆ—-algebras

Kjeld Laursen

Vol. 61 (1975), No. 2, 483–491
Abstract

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
Milestones
Received: 30 January 1975
Revised: 4 November 1975
Published: 1 December 1975
Authors
Kjeld Laursen