# Automatic continuity of linear operators by Allan M. Sinclair By Allan M. Sinclair

A number of the effects on automated continuity of intertwining operators and homomorphisms that have been acquired among 1960 and 1973 are the following accrued jointly to supply a close dialogue of the topic. The e-book should be liked through graduate scholars of useful research who have already got a very good starting place during this and within the idea of Banach algebras.

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Example text

7). By the construction of A(F), it follows that OA(F) is contained in Y(F). Theorem 2. 3 now shows that there are only a finite number of discontinuity points. If j is not a dis38 continuity point of 0 in S2, then the map A''9(X)/Y({j}) : a- 9(a) +Y({j}) is continuous. By Lemma 6. 8 the map A - X : a I- 6(a)xj = axj is continuous. This gives a contradiction. 6(a) Hence A - X : a I- 6(a)x is continuous for each x in X, and is a continuous linear operator on X for each a in A. The uniform boundedness theorem implies that 6 is continuous, and the proof is complete.

The first method is again an elementary annihilation technique (Lemma 8. 3). The second depends on an algebraic theorem for extending derivations from a field, though Dales's original proof  was a direct extension argument. We begin by giving a construction of a discontinuous monomorphism from a Banach algebra given that there is a discontinuous derivation from the algebra. This result then provides two methods of constructing discontinuous homomorphisms from the two results on discontinuous derivations that follow.

Containing no other points of F. Then ER(Uj)S = ER(Ui)B + K. because ER(Ui)ER( {A }) is equal to ER(JA }) if k = j and is zero if k * j. Since ER(Ui)S is disconi tinuous and ER(U ,)B is continuous, K. is discontinuous. I) since (R - AjI)ER(I A. )) = 0. Therefore A. is an j eigenvalue of R because Kj is non-zero. I)Y = { 0 1. If (T - AjI)X were of finite codimension in X, j then it would be closed in X by Lemma 3. 3; hence K. would be continuous on X contradicting the discontinuity of KY Thus each Aj is a critical eigenvalue of (T, R), and the proof is complete.