Proceedings of the Japan Academy, Series A, Mathematical Sciences

Borel structures coming from various topologies on $\mathbf{B}(\mathcal{H})$

Ghorban Ali Bagheri-Bardi and Minoo Khosheghbal-Ghorabayi

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Abstract

Although there exist different types of (well-known) locally convex topologies on $\mathbf{B}(\mathcal{H})$, the notion of measurability on the set of operator valued functions $f:\Omega\to \mathbf{B}(\mathcal{H})$ is unique when $\mathcal{H}$ is separable (see [1]). In this current discussion we observe that unlike the separable case, in the non-separable case we have to face different types of measurability. Moreover the algebraic operations “addition and product” are not compatible with the set of operator valued measurable functions.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci. Volume 93, Number 2 (2017), 7-11.

Dates
First available in Project Euclid: 1 February 2017

Permanent link to this document
http://projecteuclid.org/euclid.pja/1485918017

Digital Object Identifier
doi:10.3792/pjaa.93.7

Mathematical Reviews number (MathSciNet)
MR3604020

Subjects
Primary: 46L10: General theory of von Neumann algebras 47A56: Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones)
Secondary: 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05] 28A20: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence

Keywords
von Neumann algebras operator valued functions $\sigma$-algebras measurability

Citation

Bagheri-Bardi, Ghorban Ali; Khosheghbal-Ghorabayi, Minoo. Borel structures coming from various topologies on $\mathbf{B}(\mathcal{H})$. Proc. Japan Acad. Ser. A Math. Sci. 93 (2017), no. 2, 7--11. doi:10.3792/pjaa.93.7. http://projecteuclid.org/euclid.pja/1485918017.


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References

  • G. A. Bagheri-Bardi, Operator-valued measurable functions, Bull. Belg. Math. Soc. Simon Stevin 22 (2015), no. 1, 159–163.
  • V. I. Bogachev, Measure theory. Vol. I, Springer, Berlin, 2007.
  • M. Takesaki, Theory of operator algebras. I, Springer, New York, 1979.