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Strong sequential completeness of the natural domain of a conditional expectation operator in Riesz spaces


Authors: Wen-Chi Kuo, David F. Rodda and Bruce A. Watson
Journal: Proc. Amer. Math. Soc. 147 (2019), 1597-1603
MSC (2010): Primary 46B40, 60F15, 60F25
DOI: https://doi.org/10.1090/proc/14341
Published electronically: December 12, 2018
MathSciNet review: 3910424
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Abstract: Strong convergence and convergence in probability were generalized to the setting of a Riesz space with conditional expectation operator, $T$, in [Positivity 19 (2015), pp. 647–657] as $T$-strong convergence and convergence in $T$-conditional probability, respectively. Generalized $L^{p}$ spaces for the cases of $p=1,2,\infty$ were discussed in the setting of Riesz spaces as $\mathcal {L}^{p}(T)$ spaces in [Positivity 14 (2010), pp. 859–875]. An $R(T)$ valued norm, for the cases of $p=1,\infty ,$ was introduced on these spaces in [J. Math. Anal. Appl. 456 (2017), pp. 992–1004], where it was also shown that $R(T)$ is a universally complete $f$-algebra and that these spaces are $R(T)$-modules. In [J. Math. Anal. Appl. 447 (2017), pp. 798–816] functional calculus was used to consider $\mathcal {L}^{p}(T)$ for $p\in (1,\infty )$. In this paper we prove the strong sequential completeness of the space $\mathcal {L}^{1}(T)$, the natural domain of the conditional expectation operator $T$, and the strong completeness of $\mathcal {L}^{\infty }(T)$.


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Additional Information

Wen-Chi Kuo
Affiliation: School of Mathematics, University of the Witwatersrand, Private Bag 3, P O WITS 2050, South Africa
MR Author ID: 744819

David F. Rodda
Affiliation: School of Mathematics, University of the Witwatersrand, Private Bag 3, P O WITS 2050, South Africa

Bruce A. Watson
Affiliation: School of Mathematics, University of the Witwatersrand, Private Bag 3, P O WITS 2050, South Africa
MR Author ID: 649582
ORCID: 0000-0003-2403-1752

Keywords: Strong completeness, Riesz spaces, conditional expectation operators
Received by editor(s): February 22, 2018
Received by editor(s) in revised form: July 25, 2018, and July 26, 2018
Published electronically: December 12, 2018
Additional Notes: The first author was supported in part by National Research Foundation of South Africa grant no.Β CSUR160503163733.
The second author was supported in part by National Research Foundation of South Africa grant no.Β 110943.
The third author was supported in part by the Centre for Applicable Analysis and Number Theory and by National Research Foundation of South Africa grant IFR170214222646 with grant no. 109289.
Communicated by: Stephen Dilworth
Article copyright: © Copyright 2018 American Mathematical Society