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Infinitely divisible completely positive mappings
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  • Published: October 1988

Infinitely divisible completely positive mappings

  • M. Fannes1,2 &
  • J. Quaegebeur3 

Probability Theory and Related Fields volume 79, pages 369–403 (1988)Cite this article

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Summary

We generalise the theory of infinitely divisible positive definite functions f:G→ℂ on a group G to a theory of infinite divisibility for completely positive mappings Φ: G→ℬ(ℋ) taking values in the algebra of bounded operators on some Hilbert space ℋ.

We prove a structure theorem for normalised infinitely divisible completely positive mappings Φ which shows that the mapping Φ, its Stinespring representation and its Stinespring isometry are of type S (in the sense of Guichardet [Gui]). Furthermore, we prove that a completely positive mapping is infinitely divisible if and only if it is the exponential (as defined in this paper) of a hermitian conditionally completely positive mapping.

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Author information

Authors and Affiliations

  1. Bevoegdverklaard Navorser N.F.W.O., Belgium

    M. Fannes

  2. Instituut Theoretische Fysica, Universiteit Leuven, B-3030, Leuven, Belgium

    M. Fannes

  3. Department Wiskunde, Katholieke Universiteit Leuven, Celestignenlaan 200 B, B-3030, Leuven, Belgium

    J. Quaegebeur

Authors
  1. M. Fannes
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  2. J. Quaegebeur
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Fannes, M., Quaegebeur, J. Infinitely divisible completely positive mappings. Probab. Th. Rel. Fields 79, 369–403 (1988). https://doi.org/10.1007/BF00342232

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  • Received: 01 April 1987

  • Revised: 02 February 1988

  • Issue Date: October 1988

  • DOI: https://doi.org/10.1007/BF00342232

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Keywords

  • Hilbert Space
  • Stochastic Process
  • Probability Theory
  • Statistical Theory
  • Bounded Operator
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