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On Two Equivalent Dilation Theorems in VH-Spaces

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Abstract

We prove that a generalized version, essentially obtained by R.M. Loynes, of the B. Sz.-Nagy’s Dilation Theorem for \({\mathcal{B}^*(\mathcal{H})}\) -valued (here \({\mathcal{H}}\) is a VH-space in the sense of Loynes) positive semidefinite maps on *-semigroups is equivalent with a generalized version of the W.F. Stinespring’s Dilation Theorem for \({\mathcal{B}^*(\mathcal{H})}\) -valued completely positive linear maps on B *-algebras. This equivalence result is a generalization of a theorem of F.H. Szafraniec, originally proved for the case of operator valued maps (that is, when \({\mathcal{H}}\) is a Hilbert space).

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Author notes

  1. Baris Evren Ugurcan

    Present address: Department of Mathematics, Cornell University, 310 Malott Hall, Ithaca, NY, 14853-4201, USA

Authors and Affiliations

  1. Department of Mathematics, Bilkent University, Bilkent, 06800, Ankara, Turkey

    Aurelian Gheondea & Baris Evren Ugurcan

  2. Institutul de Matematică al Academiei Române, C.P. 1-764, 014700, Bucharest, Romania

    Aurelian Gheondea

Authors
  1. Aurelian Gheondea
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  2. Baris Evren Ugurcan
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Correspondence to Aurelian Gheondea.

Additional information

Communicated by Guest Editors L. Littlejohn and J. Stochel.

Dedicated to F.H. Szafraniec on the occasion of his 70th anniversary.

A. Gheondea acknowledges support of CNCSIS—Program Idei.

B. E. Ugurcan was supported by TÜBİTAK-BİDEB 2210—Yurtiçi Yüksek Lisans Burs Programı.

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Gheondea, A., Ugurcan, B.E. On Two Equivalent Dilation Theorems in VH-Spaces. Complex Anal. Oper. Theory 6, 625–650 (2012). https://doi.org/10.1007/s11785-011-0191-9

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