Automatic continuity of linear operators by Allan M. Sinclair

By Allan M. Sinclair

The various effects on computerized continuity of intertwining operators and homomorphisms that have been got among 1960 and 1973 are right here accrued jointly to supply an in depth dialogue of the topic. The ebook could be liked through graduate scholars of practical research who have already got a great origin during this and within the concept of Banach algebras.

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Remarks. Various versions of Theorem 9. 3 occur in [24], [78], [117]. Lemma 9. 1 is essentially a special case of Bade and Curtis [7, Theorem 2. 1]. The results of Sections 9, 10, and 12 do not depend on Lemma i. 6 or Theorem 2. 3. Something similar to Lemma 9. 1 may be deduced from Lemma 1. 6 but the only proof I know is harder than that 57 This suggests that 1. 6 (or 2. 3) can be strengthened, or that these are the wrong technical results. of 9. 1. 10. Homomorphisms from C(62) We begin this section with Kaplansky's Theorem on monomorphisms from C(f2) into a Banach algebra [72], [103].

Corollary 6. 12 deals with automorphisms. If * is an involution on A, let 11 II 40 I 11 I I 1 = II x* II . Then I- 1 is a complete algebra norm on A, so is equivalent to II, and hence * is continuous. 14. Remarks. The previous results on the continuity of homomorphisms were obtained by studying the global properties of 0, and their relation to irreducible representations of the algebra. Different results may be obtained by considering the detailed structure of the spectra of individual elements in the separating space; these methods are local.

N Pm is isomorphic to A/P1 ®... :n+1

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