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By Allan M. Sinclair

Many of the effects on automated continuity of intertwining operators and homomorphisms that have been acquired among 1960 and 1973 are right here accumulated jointly to supply a close dialogue of the topic. The publication could be favored through graduate scholars of sensible research who have already got a very good origin during this and within the idea of Banach algebras.

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Lemma. (a) If J is a maximal modular left ideal in A, then with the multiplication a. (b + J) = ab + J for all a, b in A the linear space A/J is an irreducible A-module. (b) If X is an irreducible A-module, then for each non-zero x in X the set J = { a e A : ax = 0 1 is a maximal modular left ideal in A. Proof. (a) Since a(x + J) = ax + J for all a and x in A, there is a one-to-one correspondence between left ideals K containing J and submodules of A/J given by K - K/J. Thus there are no proper submodules of A/J, i.

Letting a = ba' completes the induction, and the proof of the lemma. 7. Theorem. Let X be an irreducible module over a Banach algebra A. If x1, ... , xn are linearly independent in X, and y1, ... , yn are in X, then there is an a in A such that axj = y] for j = 1, ... , n. If A is unital, if 1. x = x for all x in X, and if y1, ... ' yn are linearly independent, then a can be chosen to be 36 invertible. Proof. xj * 0 by Lemma 6. 6. Because X is an irreducible A-module there is a c. x. = y.. b.. This completes the first part of the proof.

Thus M1 ®... ® Mm is isomorphic to A/P1 ®... ® A/PIn . Also nip j:1Sj:5n) +n{Pj:n+1 :5 j

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