Vol. 43, No. 1, 1972

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The non-conjugacy of certain algebras of operators

Julien O. Hennefeld

Vol. 43 (1972), No. 1, 111–113

Let E be a Banach space and B(E) be the space of all bounded linear operators on E. It was shown by Schatten, that if E is a conjugate space then B(E) is isometrically isomorphic to a conjugate space. The fact that for an arbitrary Banach space, the unit ball of B(E) has extreme points suggests that B(E) might always be a conjugate space. In this paper it is proved that if E has an unconditional basis and is not isomorphic to a conjugate space, then B(E) is not isomorphic to a conjugate space. An even stronger result is proved.

Furthermore, it is shown that if E has an unconditional basis or a complemented subspace with an unconditional basis, then the space of all compact linear operators on E is not isomorphic to a conjugate space.

Mathematical Subject Classification
Primary: 46L20
Received: 17 August 1971
Revised: 1 February 1972
Published: 1 October 1972
Julien O. Hennefeld