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Journal of Operator Theory

Volume 89, Issue 2, Spring 2023  pp. 361-427.

Multiple operator integrals in non-separable von Neumann algebras

Authors:  Evangelos A. Nikitopoulos
Author institution: Department of Mathematics, University of California San Diego, La Jolla, CA 92093-0112, U.S.A.

Summary:  A multiple operator integral (MOI) is an indispensable tool in several branches of noncommutative analysis. However, there are substantial technical issues with the existing literature on the ``separation of variables'' approach to defining MOIs, especially when the underlying Hilbert spaces are not separable. In this paper, we provide a detailed development of this approach in a very general setting that resolves existing technical issues. Along the way, we characterize several kinds of ``weak'' operator valued integrals in terms of easily checkable conditions and prove a useful Minkowski-type integral inequality for maps with values in a semifinite von Neumann algebra.

Keywords:  multiple operator integral, operator valued integral, Gel'fand--Pettis integral, integral projective tensor product, Minkowski inequality for operator valued integrals, non-separable Hilbert space

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