On $L\sp{1}$ isomorphisms

Michael Cambern
1980 Proceedings of the American Mathematical Society  
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
more » ... we use this theorem to prove an analogous result for Ll spaces. (Concerning the terminology "regular set isomorphism" as it is used in this paper, the reader is referred to [6].) Theorem. Let (Xx, 2" ¡xx) and (X2, 22, ft2) be o-finite measure spaces. If there exists an isomorphism T of Ll(Xx, 2" /x,) onto L{(X2, 22, ¡i2) satisfying \\T\\ ||F~'|| < 2, then there exists a regular set isomorphism $ of (Xx, 2" fix) onto (X2, 22, ju2).
doi:10.1090/s0002-9939-1980-0550500-6 fatcat:5rzk4vcc7zdrdiuyrleferzfly