Properties of Measurable Operators Associated with a von Neumann Algebra

Cong Cong SHEN, Li Ning JIANG, Li Guang WANG

Acta Mathematica Sinica, Chinese Series ›› 2019, Vol. 62 ›› Issue (2) : 293-302.

PDF(462 KB)
PDF(462 KB)
Acta Mathematica Sinica, Chinese Series ›› 2019, Vol. 62 ›› Issue (2) : 293-302. DOI: 10.12386/A2019sxxb0028

Properties of Measurable Operators Associated with a von Neumann Algebra

  • Cong Cong SHEN1,2, Li Ning JIANG2, Li Guang WANG3
Author information +
History +

Abstract

In this article, some properties of measurable operators associated with a von Neumann algebra are considered. The concept of step operator is defined and it is proved that any positive measurable operator can be strongly approximated by some step operators on its domain, which means that any positive measurable operator can be strongly approximated by some projections on its domain. In addition, the measurability of composition operator of measurable operator and bounded operator is discussed.

Cite this article

Download Citations
Cong Cong SHEN, Li Ning JIANG, Li Guang WANG. Properties of Measurable Operators Associated with a von Neumann Algebra. Acta Mathematica Sinica, Chinese Series, 2019, 62(2): 293-302 https://doi.org/10.12386/A2019sxxb0028

References

[1] Christensen E., Wang L., Von Neumann algebras as complemented subspaces of B(H), International Journal of Mathematics, 2014, 25(11): 1450107.
[2] Conway J. B., A Course in Operator Theory, American Mathematical Society, Rhode Island, 2000.
[3] Douglas R. G., Banach Algebra Techniques in Operator Theory, Springer, Berlin, 1998.
[4] Guo M., Real Variable Function and Functional Analysis, Peking University Press, Beijing, 2005(in Chinese).
[5] Jiang L., Ma Z., Closed subspaces and some basic topological properties of noncommutative orlicz spaces, Proceedings–Mathematical Sciences, 2016, 127(3): 1–12.
[6] Jones V. F. R., Subfactors and Knots, American Mathmatical Society, Rhode Island, 1991.
[7] Kadison R. V., Ringrose J. R., Fundamentals of the Theory of Operator Algebras (Volume I: Elementary Theory), Academic Press, San Diego, California, 1983.
[8] Kadison R. V., Ringrose J. R., Fundamentals of the Theory of Operator Algebras (Volume Ⅱ: Advanced Theory), Academic Press. INC, London, 1986.
[9] Li Q., Shen J., Shi R., et al., Perturbations of self-adjoint operators in semifinite von Neumann algebras: Kato-Rosenblum theorem, J. Functional Analysis,https://doi.org/10.1016/j.jfa.2018.04.006.
[10] Ma X., Hadwin D., Wang L., On representations of M2(C)*M2(C), J. Math. Anal. Appl., 2010, 362(1): 107–113.
[11] Meng Q., The Weak Haagerup property for C*-algebras, Ann. Funct. Anal., 2017, 8(4): 502–511.
[12] Meng Q., Weak Haagerup property of W*-crossed Products, Bull. Aust. Math. Soc., 2018, 97(1): 119–126.
[13] Murphy G. J., C*-algebras and Operator Theory, Academic Press, London, 1990.
[14] Segal I. E., Irreducible representations of operator algebras, Bulletin of the American Mathematical Society, 1947, 53(2): 73–88.
[15] Shen C., Jiang L., Egoroff's theorem in measurable operators space associated a von Neumann algebra, Indian Journal of Pure and Applied Mathematics (in press).
[16] Wang L., On the properties of some sets of von Neumann algebras under perturbation, Sci. China Math., 2015, 58(8): 1707–1714.
[17] Wang L., Yuan W., A new class of Kadison-Singer algebras, Exposition Math., 2011, 29(1): 126–132.
[18] Webster C., On unbounded operators affiliated with C*-algebras, Journal of Operator Theory, 2004, 2(2): 237–244.
[19] Wei C., Wang L., Hereditary subalgebras and comparisons of positive elements, Sci. China Math., 2010, 53: 1565–1570.
[20] Wei C., Wang L., Isomorphism of extensions of C(T2), Sci. China Math., 2011, 54: 281–286.
[21] Woronowicz S. L., Unbounded elements affiliated with C*-algebras and non-compact quantum groups, Commun. Math. Phys., 1991, 136: 399–432.
[22] Wu J., Wu W., Wang L., On Similarity Degrees of finite von Neumann algebras, Taiwanese Journal of Mathematics, 2012, 16(6): 2275–2287.
[23] Xu Q., Turdebek N. B., Chen Z., Introduction to Operator Algebras and Noncommutative Lp Spaces (in Chinese), Science Press, Beijing, 2010.

PDF(462 KB)

115

Accesses

0

Citation

Detail

Sections
Recommended

/