Strong completeness of a class of L2-type Riesz spaces

TL;DR

This paper proves the strong completeness of L2(T) spaces in Riesz spaces with a conditional expectation operator, extending the theory of convergence and duality in this mathematical framework.

Contribution

It establishes the T-strong completeness of L2(T) spaces and introduces a Riesz-Fischer type theorem with duality relative to the T-strong dual.

Findings

L2(T) is T-strong complete.

A Riesz-Fischer type theorem is developed.

T is a weak order unit for the T-strong dual.

Abstract

Strong convergence and convergence in probability were generalized to the setting of a Riesz space with conditional expectation operator, T, in [Y. Azouzi, W.-C. Kuo, K. Ramdane, B. A. Watson, Convergence in Riesz spaces with conditional expectation operators, Positivity, 19 (2015), 647-657] as T-strong convergence and convergence in T- conditional probability, respectively. Generalized Lp spaces for the cases of p = 1; 2;1, were discussed in the setting of Riesz spaces as Lp(T) spaces in [C. C. A. Labuschagne, B. A. Watson, Discrete stochastic integration in Riesz spaces, Positivity, 14 (2010), 859-875]. An R(T) valued norm, for the cases of p = 1;1; was introduced on these spaces in [W. Kuo, M. Rogans, B.A. Watson, Mixing processes in Riesz spaces, Journal of Mathematical Analysis and Application, 456 (2017), 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 [Y. Azouzi, M. Trabelsi, Lp-spaces with respect to conditional expectation on Riesz spaces, Journal of Mathematical Analysis and Application, 447 (2017), 798-816] functional calculus was used to consider Lp(T) for p 2 (1;1). The strong sequential completeness of the space L1(T), the natural domain of the conditional expectation operator T, and the strong completeness of L1(T) was established in [W.-C. Kuo, D. Rodda, B. A. Watson, Sequential strong completeness of the natural domain of Riesz space conditional expectation operators, Proc. AMS, 147 (2019), 1597{1603]. In the current work the T-strong completeness of L2(T) is established along with a Riesz- Fischer type theorem where the duality is with respect to the T-strong dual. It is also shown that the conditional expectation operator T is a weak order unit for the T-strong dual.