Abstract
In this paper, motivated by Bhat et al. (Trans. Amer. Math. Soc. 370 (2018) 2605–2637), we determine all continuous roots of the vacuum unit in the time ordered product system \(\mathrm {I}\!\mathrm {\Gamma }^{\otimes }(F)\), where F is a two-sided Hilbert module over the \(C^*\)-algebra \({\mathcal {B}}\) of all bounded operators acting on a Hilbert space of finite dimension. Afterwards, we prove that the index of that product system and the Hilbert \({\mathcal {B}}-{\mathcal {B}}\) module of all continuous roots of the vacuum unit are isomorphic as Hilbert two-sided modules.
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References
Arveson W, Continous analogues of Fock space, Mem.Amer. Math. Soc. 80 (1989) no. 409, \(iv+66\) pp.
Arveson W, Noncommutative Dynamics and E-Semigroups (2003) (Springer)
Bakić D and Guljaš B, Hilbert \(C^*\)-modules over \(C^*\)-algebras of compact operators, Acta. Sci. Math. (Szeged) 68 (2002) 249–269
Barreto S D, Bhat B V R, Liebscher V and Skeide M, Type I product systems of Hilbert modules, J. Funct. Anal. 212 (2004) 121–181
Bhat B V R, Liebscher V and Skeide M, Subsystems of Fock need not be Fock: Spatial CP-semigroups, Proc. Amer. Math. Soc. 138 (2010) 2443–2456
Bhat B V R, Lindsay M and Mukherjee M, Additive units of product systems, Trans. Amer. Math. Soc. 370 (2018) 2605–2637
Bhat B V R and Skeide M, Tensor product systems of Hilbert modules and dilations of completely positive semigroups, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 (2000) 519–575
Frank M and Paulsen V I, Injective and projective Hilbert \(C^*\)-modules and \(C^*\)-algebras of compact operators, arXiv:math/0611349v2 [math.OA] (2008)
Kečkić D J, Vujošević B, On the index of product systems of Hilbert modules, Filomat. 29(5) (2015) 1093–1111
Lance E C, Hilbert \(C^*\)-Modules: A toolkit for operator algebraists (1995) (Cambridge University Press)
Magajna B, Hilbert \(C^*\)-modules in which all closed submodules are complemented, Proc. Amer. Math. Soc. 125(3) (1997) 849–852, S 0002-9939(97)03551-X
Manuilov V M and Troitsky E V, Hilbert \(C^*\)-Modules, American Mathematical Society (2005)
Skeide M, Dilation theory and continuous tensor product systems of Hilbert modules, PQ–QP: Quantum Probability and White Noise Analysis XV (2003) (World Scientific)
Skeide M, Hilbert modules and application in quantum probability, Habilitationsschrift, Cottbus (2001)
Skeide M, The index of (white) noises and their product systems, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2006) 617–655
Vujošević B, Additive units of product system of Hilbert modules, Int. J. Anal. Appl., 10(2) (2016) 71–76
Vujošević B, The index of product systems of Hilbert modules: two equivalent definitions, Publ. de l’Institute Mathematique 97(111) (2015) 49–56
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This research was supported by the Serbian Ministry of Education, Science and Technological Development through Faculty of Mathematics, University of Belgrade.
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Communicated by B V Rajarama Bhat.
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Vujošević, B. On the index and roots of time ordered product systems. Proc Math Sci 132, 1 (2022). https://doi.org/10.1007/s12044-021-00647-2
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DOI: https://doi.org/10.1007/s12044-021-00647-2
Keywords
- Product systems
- Hilbert \(C^*\)-modules
- index
- time ordered product systems
- additive units of product systems