## Advances in Operator Theory

### A trick for investigation of near-martingales in quantum probability spaces

#### Abstract

‎‎‎We introduce near-martingales in the setting of quantum probability spaces and present a trick for investigating some of their properties‎. ‎For instance‎, ‎we give a near-martingale analogous result of the fact that the space of all bounded $L^p$-martingales‎, ‎equipped with the norm $\|\cdot\|_p$‎, ‎is isometric to $L^p(\mathfrak{M})$ for $p>1$‎. ‎We also present Doob and Riesz decompositions for the near-submartingale and provide Gundy's decomposition for $L^1$-bounded near-martingales‎. ‎In addition‎, ‎the interrelation between near-martingales and the instantly independence is studied‎.

#### Article information

Source
Adv. Oper. Theory, Volume 4, Number 4 (2019), 784-792.

Dates
Received: 18 February 2019
Accepted: 1 April 2019
First available in Project Euclid: 15 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.aot/1557885626

Digital Object Identifier
doi:10.15352/aot.1903-1484

Mathematical Reviews number (MathSciNet)
MR3949975

Zentralblatt MATH identifier
07064105

Subjects
Primary: 46L53: Noncommutative probability and statistics
Secondary: 46L10‎ ‎60E15‎ ‎47A30

#### Citation

Sadeghi, Ghadir; Talebi, Ali. A trick for investigation of near-martingales in quantum probability spaces. Adv. Oper. Theory 4 (2019), no. 4, 784--792. doi:10.15352/aot.1903-1484. https://projecteuclid.org/euclid.aot/1557885626

#### References

• C. Barnett, Supermartingales on semi-finite von Neumann algebras, J. London Math. Soc. 24 (1981), 175–181.
• D. Bartl, P. Cheridito, M. Kupper, and L. Tangpi, Duality for increasing convex functionals with countably many marginal constraints, Banach J. Math. Anal. 11 (2017), no. 1, 72–89.
• B. de Pagter, Noncommutative Banach function spaces, Positivity, 197–227, Trends Math., Birkhüser, Basel, 2007.
• M. Junge and Q. Xu, Noncommutative Burkholder/Rosenthal inequalities. II. Applications, Israel J. Math. 167 (2008), 227–282.
• H.-H. Kuo, The Itô calculus and white noise theory: a brief survey toward general stochastic integration, Commun. Stoch. Anal. 8 (2014), no. 1, 111–139.
• H.-H. Kuo, A. Sae-Tang, and B. Szozda, A stochastic integral for adapted and instantly independent stochastic processes, in “Advances in Statistics, Probability and Actuarial Science" Vol. I, Stochastic Processes, Finance and Control: A Festschrift in Honour of Robert J. Elliott (eds.: Cohen, S., Madan, D., Siu, T. and Yang, H.), World Scientific, (2012), 53–71.
• H.-H. Kuo and K. Saitô, Doob's decomposition theorem for near-submartigales, Comm. Stoch. Anal. 9 (2015), no. 4, 467–476.
• J. Parcet and N. Randrianantoanina, Gundy's decomposition for non-commutative martingales and applications, Proc. London Math. Soc. (3) 93 (2006), no. 1, 227–252.
• L. E. Persson, G. Tephnadze, and P. Wall, On an approximation of 2-dimensional Walsh-Fourier series in martingale Hardy spaces, Ann. Funct. Anal. 9 (2018), no. 1, 137–150.
• A. Talebi, M. S. Moslehian and Gh. Sadeghi, Noncommutative Blackwell–Ross martingale inequality, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 21 (2018), no. 1, 1850005, 9 pp.
• A. Talebi, M. S. Moslehian and Gh. Sadeghi, Etemadi and Kolmogorov inequalities in noncommutative probability spaces, Michigan Math. J. 68 (2019), no. 1, 57–69.
• Q. Xu, Operator spaces and noncommutative $L_p$, Lectures in the Summer School on Banach spaces and Operator spaces, Nankai University China, 2007.