Abstract
We review the concept of a weighted noncommutative Banach function space. This concept constitutes a generalisation of the by now well-known theory of noncommutative Banach function spaces associated with a semifinite von Neumann algebra. In this review we remind the reader of the quantum statistical problem which gave birth to this concept, we investigate the extent to which a weighted theory of measurable operators agrees with the standard theory, we explore competing methods for defining such spaces, before finally describing the monotone interpolation theory of such spaces.
Dedicated to Prof Ben de Pagter on the occasion of his 65th birthday
The contribution of L. E. Labuschagne is based on research partially supported by the National Research Foundation (IPRR Grant 96128). Any opinion, findings and conclusions or recommendations expressed in this material are those of the author, and therefore the NRF does not accept any liability in regard thereto.
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References
C. Bennett, R. Sharpley, Interpolation of Operators (Academic, Boston, 1988)
P.G. Dodds, T.K.-Y. Dodds, B. de Pagter, Non-commutative Banach function spaces. Math. Z. 201, 583–597 (1989)
P.G. Dodds, T. K.-Y. Dodds, B. de Pagter, Fully symmetric operator spaces. Integral Equ. Oper. Theory 15, 942–972 (1992)
P.G. Dodds, T.K.-Y. Dodds, B. de Pagter, Noncommutative Köthe duality. Trans. Am. Math. Soc. 339, 717–750 (1993)
P.G. Dodds, F.A. Sukochev, G. Schlüchtermann, Weak compactness criteria in symmetric spaces of measurable operators. Math. Proc. Camb. Philos. Soc. 131, 363–384 (2001)
T. Fack, H. Kosaki, Generalized s-numbers of τ-measurable operators. Pac. J. Math. 123, 269–300 (1986)
U. Haagerup, M. Junge, Q. Xu, A reduction method for noncommutative L p-spaces and applications. Trans. Am. Math. Soc. 362, 2125–2165 (2010)
L.E. Labuschagne, W.A. Majewski, Maps on non-commutative Orlicz spaces. Ill. J. Math. 55, 1053–1081 (2011)
L.E. Labuschagne, W.A. Majewski, Quantum dynamics on Orlicz spaces. arXiv:1605.01210 [math-ph]
W.A. Majewski, L.E. Labuschagne, On applications of Orlicz Spaces to Statistical Physics. Ann. Henri Poincaré 15, 1197–1221 (2014)
E. Nelson, Notes on non-commutative integration. J. Funct. Anal. 15, 103–116 (1974)
G. Pistone, C. Sempi, An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one. Ann. Stat. 23, 1543–1561 (1995)
C. Steyn, An alternative approach to weighted non-commutative Banach function spaces. arXiv:1810.12753 [math.OA]
M. Takesaki, Theory of Operator Algebras, Vol I,II,III (Springer, New York, 2003)
M. Terp, L p spaces associated with von Neumann algebras. Københavs Universitet, Mathematisk Institut, Rapport No 3a (1981)
J. von Neumann, Some matrix inequalities and metrization of matrix space. Tomsk Univ. Rev. 1, 286–300 (1937)
Q. Xu, Analytic functions with values in lattices and symmetric spaces of measurable operators. Math. Proc. Camb. Philos. Soc. 109, 541–563 (1991)
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Labuschagne, L.E., Steyn, C. (2019). Weighted Noncommutative Banach Function Spaces. In: Buskes, G., et al. Positivity and Noncommutative Analysis. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-10850-2_17
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DOI: https://doi.org/10.1007/978-3-030-10850-2_17
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