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Isometries of measurable functions

Published online by Cambridge University Press:  17 April 2009

Michael Cambern
Michael Cambern
Affiliation:
Department of Mathematics, University of California, Santa Barbara, California 93106, USA.
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Abstract

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Let (X, Σ, μ) be a σ-finite measure space and denote by L∞(X, K) the Banach space of essentially bounded, measurable functions F defined on X and taking values in a separable Hilbert space K. In this article a characterization is given of the linear isometries of L∞(X, K) onto itself. It is shown that if T is such an isometry then T is of the form (T(F))(x) = U(x)(φ(F))(x), where φ is a set isomorphism of Σ onto itself, and U is a measurable operator-valued function such that U(x) is almost everywhere an isometry of K onto itself. It is a consequence of the proof given here that every isometry of L∞(X, K) is the adjoint of an isometry of L1(x, K).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 24 , Issue 1 , August 1981 , pp. 13 - 26
Copyright
Copyright © Australian Mathematical Society 1981

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