Allan M. Sinclair's Automatic continuity of linear operators PDF
By Allan M. Sinclair
A number of the effects on automated continuity of intertwining operators and homomorphisms that have been received among 1960 and 1973 are right here accumulated jointly to supply an in depth dialogue of the topic. The booklet may be preferred via graduate scholars of useful research who have already got a very good beginning during this and within the conception of Banach algebras.
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Additional info for Automatic continuity of linear operators
T, R) has no critical eigenvalue; hence r. r. iXk=Xk. (T - X I) 1X is of finite codimension in X. If k # j, then (T XP Therefore we have r (T - xjI) iX = X1 ®... ED Xj-1 ® Xj+1 ®... ® Xn is an eigenvalue of R, the linear space Xj is finite dimensional. Therefore S is zero on the direct sum of the Xk for those Xk that are not eigenvalues of R. This direct sum is a closed linear subspace of finite codimension in X. Hence S is continuous. So if A. Let X and Y be Banach spaces, and let T and R be continuous linear operators on X and Y, respectively, such that o(T) n a(R) is empty.
Lemma 1. 6 is applied in Section 11 to a homomorphism from C o(ff) into a radical Banach algebra to prove that the image of C0(*) contains no nilpotent elements, where 4' is a locally compact Hausdorff space. In Section 12 we use Theorem 9. 3 to obtain several results on the automatic continuity of homomorphisms from C*-algebras. 32 6. Isomorphisms of semisimple Banach algebras The main result of this section is the uniqueness of the norm topology on a Banach space that is an irreducible module over a Banach algebra such that algebra multiplication on the module is continuous (Johnson ) (Theorem 6.
Both Sections 6 and 7 depend on Theorem 2. 3. Section 8 contains a brief discussion of the relationship between discontinuous derivations and discontinuous automorphisms, and a proof of the existence of discontinuous derivations from the disc algebra into a Banach module over it . Sections 9 to 12 are concerned with homomorphisms from C*algebras and, in particular, from the Banach algebra of continuous complex valued functions on a compact Hausdorff space. The main technical result is Theorem 9.
Automatic continuity of linear operators by Allan M. Sinclair