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Dilation theory in finite dimensions and matrix convexity

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Abstract

We establish a finite-dimensional version of the Arveson-Stinespring dilation theorem for unital completely positive maps on operator systems. This result can be seen as a general principle to deduce finite-dimensional dilation theorems from their classical infinite-dimensional counterparts. In addition to providing unified proofs of known finite-dimensional dilation theorems, we establish finite-dimensional versions of Agler’s theorem on rational dilation on an annulus, of Berger’s dilation theorem for operators of numerical radius at most 1, and of the Putinar-Sandberg numerical range dilation theorem. As a key tool, we prove versions of Carathéodory’s and of Minkowski’s theorems for matrix convex sets.

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Acknowledgements

The authors are grateful to John McCarthy, to Michael Dritschel and to an anonymous referee for asking questions that led to Corollaries 4.8 and 4.7. Moreover, the authors thank David Sherman for bringing [38] to their attention. Finally, the authors greatly appreciate the careful reading and helpful comments of an anonymous referee.

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Authors and Affiliations

  1. Fachrichtung Mathematik, Universität des Saarlandes, 66123, Saarbrücken, Germany

    Michael Hartz

  2. School of Mathematics and Statistics, Victoria University of Wellington, PO Box 600, Wellington, 6140, New Zealand

    Martino Lupini

Authors
  1. Michael Hartz
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  2. Martino Lupini
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Correspondence to Michael Hartz.

Additional information

M. H. was partially supported by a Feodor Lynen Fellowship and by a GIF grant.

M. L. was partially supported by the NSF Grant DMS-1600186, by a Research Establishment Grant from Victoria University of Wellington, and by a Marsden Fund Fast-Start Grant from the Royal Society of New Zealand.

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Hartz, M., Lupini, M. Dilation theory in finite dimensions and matrix convexity. Isr. J. Math. 245, 39–73 (2021). https://doi.org/10.1007/s11856-021-2202-5

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  • DOI: https://doi.org/10.1007/s11856-021-2202-5

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