Hilbert von Neumann modules versus concrete von Neumann modules. (English) Zbl 1512.46037
Accardi, Luigi (ed.) et al., Infinite dimensional analysis, quantum probability and applications, QP41. Proceedings of the 41st conference, United Arab Emirates University (UAEU), Al Ain, Abu Dhabi, United Arab Emirates, virtual, March 28 – April 1, 2021. Cham: Springer. Springer Proc. Math. Stat. 390, 169-182 (2022).
Summary: Apart from presenting some new insights and results, one of our main purposes is to put some records in the development of von Neumann modules straight. The von Neumann or \(W^*\)-objects among the Hilbert \((C^*\text{-})\)modules are around since the first papers by W. L. Paschke [Trans. Am. Math. Soc. 182, 443–468 (1973; Zbl 0239.46062)] and
M. A. Rieffel [Adv. Math. 13, 176–257 (1974; Zbl 0284.46040); J. Pure Appl. Algebra 5, 51–96 (1974; Zbl 0295.46099)]
that lift Kaplansky’s setting to modules over noncommutative \(C^*\)-algebras. While the formal definition of \(W^*\)-modules is due to Baillet, Denizeau, and Havet
[M. Baillet et al., Compos. Math. 66, No. 2, 199–236 (1988; Zbl 0657.46041)], the one of von Neumann modules as strongly closed operator spaces started with
M. Skeide [Math. Proc. R. Ir. Acad. 100A, No. 1, 11–38 (2000; Zbl 0972.46038) = [19]].
It has been paired with the definition of concrete von Neumann modules in
M. Skeide [J. Oper. Theory 54, No. 1, 119–124 (2005; Zbl 1105.46039) = [23]].
It is well known that (pre-)Hilbert modules can be viewed as ternary rings of operators and in their definition of Hilbert-von Neumann modules, Bikram, Mukherjee, Srinivasan, and Sunder
[P. Bikram et al., Commun. Stoch. Anal. 6, No. 1, 49–64 (2012; Zbl 1331.46046) = [4]]
take that point of view.
We illustrate that a (suitably nondegenerate) Hilbert-von Neumann module is the same thing as a strongly full concrete von Neumann module. We also use this occasion to put some things in the papers [loc. cit. [4, 19, 23]] right. We show that the tensor product of (concrete, or not) von Neumann correspondences is, indeed, (a generalization of) the tensor product of Connes correspondences (claimed in [M. Skeide, in: Quantum stochastics and information. Statistics, filtering and control. Proceedings of the symposium quantum probability, information and control symposium, Nottingham, UK, July 15–22, 2006. Hackensack, NJ: World Scientific. 47–86 (2008; Zbl 1187.46054)]), viewed in a way quite different from [loc. cit. [4]]. We also present some new arguments that are useful even for (pre-)Hilbert modules.
For the entire collection see [Zbl 1497.46002].
We illustrate that a (suitably nondegenerate) Hilbert-von Neumann module is the same thing as a strongly full concrete von Neumann module. We also use this occasion to put some things in the papers [loc. cit. [4, 19, 23]] right. We show that the tensor product of (concrete, or not) von Neumann correspondences is, indeed, (a generalization of) the tensor product of Connes correspondences (claimed in [M. Skeide, in: Quantum stochastics and information. Statistics, filtering and control. Proceedings of the symposium quantum probability, information and control symposium, Nottingham, UK, July 15–22, 2006. Hackensack, NJ: World Scientific. 47–86 (2008; Zbl 1187.46054)]), viewed in a way quite different from [loc. cit. [4]]. We also present some new arguments that are useful even for (pre-)Hilbert modules.
For the entire collection see [Zbl 1497.46002].
MSC:
| 46L08 | \(C^*\)-modules |
Keywords:
von Neumann modules; \(W^*\)-modules; Hilbert-von Neumann modules; tensor product of von Neumann correspondencesCitations:
Zbl 0239.46062; Zbl 0284.46040; Zbl 0295.46099; Zbl 0657.46041; Zbl 0972.46038; Zbl 1105.46039; Zbl 1331.46046; Zbl 1187.46054References:
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