Download Automatic continuity of linear operators by Allan M. Sinclair PDF

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 accumulated jointly to supply a close dialogue of the topic. The publication may be liked by means of graduate scholars of practical research who have already got a very good origin during this and within the conception of Banach algebras.

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The maximality of 34 J implies that J = J . This proves the lemma. 4. Remark. (a) The above lemma is the vital link between the algebraic and topological properties which lead to the uniqueness of the complete norm topology on a semisimple Banach algebra. (b) In an algebra with identity every ideal is modular. If K is a proper modular left ideal in A with right modular identity e, then an application of Zorn's lemma gives a maximal left ideal J in A containing K and not containing e. Thus maximal modular left ideals exist in an algebra with identity.

Then the spectrum of x is a connected subset of C containing 0, and Al - x is a joint topological divisor of zero in B for each A in a(x). Proof. We may assume that A and B are unital by adjoining identities if necessary. Suppose 0 is not in a(x). Let (an) be a sequence in A with 11 an 11 -, 0 and 11 B(an) - x 11 - 0. Then choose a compact neighbourhood V of 0 in C so that 0 is not in a(x) + V. For n sufficiently large v(an) n (a(x) + V) is empty, and so v(6(an))n(a(x)+V) is empty because a(O(an)) c a(an).

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. Choose and fix a non-zero element z in X, and define II x II = inf { II b II : b E A, bz = x } for all x in X. Then as in the proof of 6. 5, X with - is a Banach space. If a is in A and x in X, II II then bz = x implies that (ab)z = ax and I I a b II < I I a I I .

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