A Banach algebra A Banach algebra A with a Banach subalgebra B and an element b Î B such that sp(A,b) is a proper subset of sp(B,b).

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Consider A(D)1 and the isometric isomorphism f\longmapsto f|T, from A(D) onto the closed subalgebra B of A = C(T) generated by 1 and inclusion z: T ® C (T is the unit circle). Then sp(B,z) = sp(A(D),z) = D and sp(A,z) = T.

Footnotes:

1Let D denote the closed unit disc {z Î C, |z| £ 1}. Suppose that A(D) denoted the set of all elements of C(D) which are analytic on the interior of D. A(D) is a closed subalgebra of C(D).

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On 22 Feb 2001, 00:15.