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Representing a Product System Representation as a Contractive Semigroup and Applications to Regular Isometric Dilations

Published online by Cambridge University Press:  20 November 2018

Orr Moshe Shalit
Orr Moshe Shalit*
Affiliation:
Department of Mathematics, Technion, Haifa, Israel e-mail: orrms@tx.technion.ac.il
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Abstract

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In this paper we propose a new technical tool for analyzing representations of Hilbert ${{C}^{*}}$- product systems. Using this tool, we give a new proof that every doubly commuting representation over ${{\mathbb{N}}^{k}}$ has a regular isometric dilation, and we also prove sufficient conditions for the existence of a regular isometric dilation of representations over more general subsemigroups of $\mathbb{R}_{+}^{k}$.

Keywords

47A2046L08
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 53 , Issue 3 , 01 September 2010 , pp. 550 - 563
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] Arveson, W., Noncommutative dynamics and E-semigroups. Springer Monographs in Mathematics, Springer-Verlag, New York, 2003.Google Scholar
[2] Lance, E. C., Hilbert C*-modules. A toolkit for operator algebraists. London Mathematical Society Lecture Note Series, 210, Cambridge University Press, Cambridge, 1995.Google Scholar
[3] Fowler, N. J., Discrete product systems of Hilbert bimodules. Pacific J. Math. 204, no. 2 (2002), 335–375. doi:10.2140/pjm.2002.204.335Google Scholar
[4] Muhly, P. and Solel, B., Tensor algebras over C*-correspondences: representations, dilations, and C*-envelopes. J. Funct. Anal. 158(1998), no. 2, 389–457. doi:10.1006/jfan.1998.3294Google Scholar
[5] Muhly, P. and Solel, B., Quantum Markov processes (Correspondences and Dilations). Internat. J. Math. 13(2002), no. 8, 863–906. doi:10.1142/S0129167X02001514Google Scholar
[6] Shalit, O. M., Dilation theorems for contractive semigroups., 2007, http://arxiv.org/abs/1004.0723v1 Google Scholar
[7] Shalit, O. M., E 0 -dilation of strongly commuting CP 0 -semigroups.. J. Funct. Anal. 255(2008), no. 1, 46–89. doi:10.1016/j.jfa.2008.04.003Google Scholar
[8] Skeide, M., Product Systems; a Survey with commutants in view. In: Quantum stochastics and information, World Sci. Publ., Hackensack, NJ, 2008.Google Scholar
[9] Solel, B., Regular dilations of representations of product systems. Math. Proc. R. Ir. Acad. 108(2008), no. 1, 89–110. doi:10.3318/PRIA.2008.108.1.89Google Scholar
[10] Sekefal’vi-Nad’, B. and Fojaş, C., Harmonic analysis of operators in Hilbert space. Izdat “Mir”, Moscow, 1970.Google Scholar