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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On $ L\sp{1}$ isomorphisms


Author: Michael Cambern
Journal: Proc. Amer. Math. Soc. 78 (1980), 227-228
MSC: Primary 46E30; Secondary 46B25
DOI: https://doi.org/10.1090/S0002-9939-1980-0550500-6
MathSciNet review: 550500
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Abstract: Let $ ({X_1},{\Sigma _1},{\mu _1})$ and $ ({X_2},{\Sigma _2},{\mu _2})$ be two $ \sigma $-finite measure spaces. We show that any isomorphism T of the Banach space $ {L^1}({X_1},{\Sigma _1},{\mu _1})$ onto the Banach space $ {L^1}({X_2},{\Sigma _2},{\mu _2})$ which satisfies $ \left\Vert T \right\Vert\;\left\Vert {{T^{ - 1}}} \right\Vert < 2$ induces a transformation of the underlying measure spaces.


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DOI: https://doi.org/10.1090/S0002-9939-1980-0550500-6
Article copyright: © Copyright 1980 American Mathematical Society

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