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On $L\sp{1}$ isomorphisms

1980
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Proceedings of the American Mathematical Society
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Let (X¡, 2(, ji,) and (X2, 22, ji^) be two o-finite measure spaces. We show that any isomorphism 7* of the Banach space L\XX, 21; /i,) onto the Banach space Ll(X2, 22> lh) which satisfies ||r|| ||r-l|| < 2 induces a transformation of the underlying measure spaces. In [1] and [2] it has been shown by D. Amir and M. Cambern that if Yx and Y2 are compact Hausdorff spaces, and if there exists an isomorphism T of C( Yx) onto C(Y2) with ||F|| ||F_1|| < 2, then Yx and Y2 are homeomorphic. In this

doi:10.1090/s0002-9939-1980-0550500-6
fatcat:5rzk4vcc7zdrdiuyrleferzfly