Download Automatic continuity of linear operators by Allan M. Sinclair 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 amassed jointly to supply an in depth dialogue of the topic. The publication might be preferred via graduate scholars of sensible research who have already got a superb beginning during this and within the concept of Banach algebras.

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Since the Pi are distinct we have k = m. Thus M1 ®... ® Mm is isomorphic to A/P1 ®... ® A/PIn . Also nip j:1Sj:5n) +n{Pj:n+1 :5 j

Theorem 2. 3 now shows that there are only a finite number of discontinuity points. If j is not a dis38 continuity point of 0 in S2, then the map A''9(X)/Y({j}) : a- 9(a) +Y({j}) is continuous. By Lemma 6. 8 the map A - X : a I- 6(a)xj = axj is continuous. This gives a contradiction. 6(a) Hence A - X : a I- 6(a)x is continuous for each x in X, and is a continuous linear operator on X for each a in A. The uniform boundedness theorem implies that 6 is continuous, and the proof is complete. An irreducible module over a Banach algebra has a unique Banach space topology such that multiplication by each element in the algebra is a continuous linear operator on the module.

Then the induced isomorphism from A/R0 to B/P0 is continuous and real linear, and so preserves the spectral radius. Since the spectral radius of x is bounded by x 11, the spectral radius of y + P is bounded by (X)-1 I < . IXI . Now the spectrum of y 0 in B is the union of the spectra of y + P in B/P as P runs over all the primitive ideals in B. With the exception of an isolated point at 1, which arises from the ideal P1, the spectrum of y lies in the open disc of radius i and centre 0. Let e be the idempotent given by the single 11 x 11 I variable analytic functional calculus corresponding to the disconnection of the spectrum of y described above so that e"(1) = 1 and e" is zero in the disc of radius z , where e" denotes the Gelfand transformation of e in the unital Banach algebra generated by y.

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