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# Automatic continuity of linear operators by Allan M. Sinclair

By Allan M. Sinclair

Many of the effects on automated continuity of intertwining operators and homomorphisms that have been bought among 1960 and 1973 are the following gathered jointly to supply an in depth dialogue of the topic. The ebook can be favored by way of graduate scholars of sensible research who have already got an excellent origin during this and within the concept of Banach algebras.

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Suppose 8* is not onto 4' 62, and let Lo be in Q\8*,b. We choose h in C(l2) such that h is 1 in a neighbourhood of A0 and h(8*4) _ {0 }, and a non-zero g in C(1) such that hg = g. Since h(6*+) = {0 } it follows that (8h)"(f) = {0 1 so that Oh is in the radical of B. From (1 - 8(h))8(g) = 0 we obtain 8(g) = 0. This contradicts 6 being a monomorphism. Hence 0*1 = 62. If f is in C(62), then IlfII = suplf(62) I = suplf(9*l) I = supl(Of)(l) I = spectral radius of Of < II Of II. This completes the proof.

Therefore hZ = Z for all non-zero h in D because D = D-1P. The that Z is D divisible. If f torsion free property also follows from the relation D = D-1P, and the proof is complete. We apply this theorem to obtain [28, Theorem 2]. There are many continuous linear operators T with the properties required in the example. Any quasinilpotent operator T on a Banach space X such that (TX) = X and T is one-to-one is a suitable candidate (see §3, [68], [1], or [28]). 8. 8. Example. Let 0 be the algebra of germs of analytic functions on the closure O of the open unit disc 0 = I IA EC : I p I < 1 and let 6) be the subalgebra of 0 of polynomials in one variable.

V n 1 (b) R = R ®... ® Rn where R. (C(ft))) _ , (c) RJRk = {0 } if j # k, (d) the restriction of P. (J({Aj}))= {0}. Proof. (i) This follows from Theorem 9. 3(i) by regarding C(Q) as a Banach C(62)-module and B as a C(62)-module by f. b = 6(f). b for all f in C(62) and b in B. (ii) Choose fl, ... , fn in C(62) so that fjfk = 0 if j * k and is equal to 1 in a neighbourhood of Xj for each j. Let D = Cf1 ® ... ® Cfn G J(F). Then D is a subalgebra of C(62) because = fj modulo J(F). Let µ0 be the restriction of 6 to D.