## Proceedings of the Japan Academy, Series A, Mathematical Sciences

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

#### 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

http://projecteuclid.org/euclid.pja/1485918017

Digital Object Identifier
doi:10.3792/pjaa.93.7

Mathematical Reviews number (MathSciNet)
MR3604020

#### 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.

#### 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.