Abstract
In this paper, we investigate the conditional expectation on the non-commutative \(H^{(r,s)}_{p}(\mathcal {A};\ell _{\infty })\) and \(H_{p}(\mathcal {A};\ell _{1})\) spaces associated with semifinite subdiagonal algebra, and prove the contractibility of the underlying conditional expectation on these spaces.
Similar content being viewed by others
References
Arveson, W.B.: Analiticity in operator algebras. Am. J. Math. 89, 578–642 (1967)
Bekjan, T.N., Xu, Quanhua: Riesz and Szegö type factorizations for noncommutative Hardy spaces. J. Oper. Theory 62(1), 215–231 (2009)
Bekjan, T.N.: Noncommutative Hardy space associated with semi-finite subdiagonal algebras. J. Math. Anal. Appl. 429, 1347–1369 (2015)
Bekjan, T.N.: Noncommutative symmetric Hardy spaces. Integral Equ. Oper. Theory 81, 191–212 (2015)
Bekjan, T.N., Sageman, B.K.: A property of conditional expectation. Positivity 22(5), 1359–1369 (2018)
Bekjan, T.N., Tulenov, K., Dauitbek, D.: The noncommutative \(H_p^{(r, s)}(\cal{A}; \ell _{\infty })\) and \(H_p(\cal{A};\ell _{1})\) spaces. Positivity 19(4), 877–891 (2015)
Blecher, D.P., Labuschagne, L.E.: Applications of the Fuglede–Kadison determinant: Szegö’s theorem and outers for noncommutative \(H^{p}\). Trans. Am. Math. Soc. 360, 6131–6147 (2008)
Blecher, D.P., Labuschagne, L.E.: Characterizations of noncommutative \(H^{\infty }\). Integral Equ. Oper. Theory 56, 301–321 (2006)
Junge, M.: Doob’s inequality for non-commutative martingales. J. Reine Angew. Math. 549, 149–190 (2002)
Defant, A., Junge, M.: Maximal theorems of Menchoff-Rademacher type in non-commutative \(L_{q}\)-spaces. J. Funct. Anal. 206, 322–355 (2004)
Dirksen, S.: Noncommutative boyd interpolation theorems. Trans. Am. Math. Soc. 367(6), 4079–4110 (2015)
Exel, R.: Maximal subdiagonal algebras. Am. J. Math. 110, 775–782 (1988)
Ji, G.: Maximality of semifinite subdiagonal algebras. J. Shaanxi Normal Univ. Nat. Sci. Ed. 28(1), 15–17 (2000)
Junge, M., Xu, Q.: Noncommutative maximal ergodic theorems. J. Am. Math. Soc. 20, 385–439 (2007)
Labuschagne, L.E.: A noncommutative Szegö theorem for subdiagonal subalgebras of von Neumann algebras. Proc. Am. Math. Soc. 133, 3643–3646 (2005)
Marsalli, M., West, G.: Noncommutative \(H^{p}\) spaces. J. Oper. Theory 40, 339–355 (1998)
Pisier, G.: Interpolation between \(H^{p}\) spaces and non-commutative generalizations I. Pac. J. Math. 155, 341–368 (1992)
Pisier, G.: Non-commutative vector valued \(L^p\)-spaces and completely p-summing maps. Astérisque 247 (1998). 5189, 667-698 (1997)
Pisier, G., Xu, Q.: Noncommutative \(L^p\)-spaces. Handb. Geom. Banach Spaces 2, 1459–1517 (2003)
Saito, K.S.: A note on invariant subspaces for finite maximal subdiagonal algebras. Proc. Am. Math. Soc. 77, 348–352 (1979)
Sukochev, F., Tulenov, K., Zanin, D.: Nehari-type theorem for non-commutative Hardy spaces. J. Geom. Anal. 27(3), 1789–1802 (1979)
Tulenov, K.S.: Noncommutative vector-valued symmetric Hardy spaces. Russ. Math. 59(11), 74–79 (2015)
Tulenov, K.: Some properties of the noncommutative \(H_p^{(r, s)}(\cal{A}; \ell _{\infty })\) and \(H_p(\cal{A};\ell _1)\) spaces. AIP Conf. Proc. 1676, 020093 (2015)
Acknowledgements
The work was partially supported by the grant (no. AP08052004 and no. AP08051978) of the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan. We thank the anonymous referee for reading the paper carefully.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Yong Jiao.
Rights and permissions
About this article
Cite this article
Dauitbek, D., Tulenov, K. Conditional expectation on non-commutative \(H^{(r,s)}_{p}(\mathcal {A};\ell _{\infty })\) and \(H_{p}(\mathcal {A};\ell _{1})\) spaces: semifinite case. Ann. Funct. Anal. 11, 617–625 (2020). https://doi.org/10.1007/s43034-019-00042-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s43034-019-00042-z
Keywords
- Semifinite von Neumann algebra
- Semifinite subdiagonal algebra
- Non-commutative Hardy spaces
- Conditional expectation