## Annals of Functional Analysis

### A bounded transform approach to self-adjoint operators: Functional calculus and affiliated von Neumann algebras

#### Abstract

Spectral theory and functional calculus for unbounded self-adjoint operators on a Hilbert space are usually treated through von Neumann’s Cayley transform. Using ideas of Woronowicz, we redevelop this theory from the point of view of multiplier algebras and the so-called bounded transform (which establishes a bijective correspondence between closed operators and pure contractions). This also leads to a simple account of the affiliation relation between von Neumann algebras and self-adjoint operators.

#### Article information

Source
Ann. Funct. Anal., Volume 7, Number 3 (2016), 411-420.

Dates
Received: 20 July 2015
Accepted: 10 December 2015
First available in Project Euclid: 17 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.afa/1466164866

Digital Object Identifier
doi:10.1215/20088752-3605384

Mathematical Reviews number (MathSciNet)
MR3513125

Zentralblatt MATH identifier
1351.47018

Subjects
Primary: 47B25: Symmetric and selfadjoint operators (unbounded)
Secondary: 46L10: General theory of von Neumann algebras

#### Citation

Budde, Christian; Landsman, Klaas. A bounded transform approach to self-adjoint operators: Functional calculus and affiliated von Neumann algebras. Ann. Funct. Anal. 7 (2016), no. 3, 411--420. doi:10.1215/20088752-3605384. https://projecteuclid.org/euclid.afa/1466164866

#### References

• [1] S. Baaj and P. Julg, Théorie bivariante de Kasparov et opérateurs non bornés dans les $C^{*}$-modules hilbertiens, C. R. Math. Acad. Sci. Paris Sér. ı Math. 296 (1983), no. 21, 875–878.
• [2] C. Budde, Operator algebras and unbounded self-adjoint operators, M.Sc. dissertation, Radboud University, Nijmegen, the Netherlands, 2015, www.math.ru.nl/~landsman/Budde.pdf (accessed 9 May 2016).
• [3] W. E. Kaufman, Closed operators and pure contractions in Hilbert space, Proc. Amer. Math. Soc. 87 (1983), no. 1, 83–87.
• [4] J. J. Koliha, On Kaufman’s theorem, J. Math. Anal. Appl. 411 (2014), no. 2, 688–692.
• [5] E. C. Lance, Hilbert $C^{*}$-modules: A Toolkit for Operator Algebraists, London Math. Soc. Lecture Note Ser. 210, Cambridge Univ. Press, Cambridge, 1995.
• [6] E. Nelson, Analytic vectors, Ann. of Math. (2) 70 (1959), 572–614.
• [7] J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Unveränderter Nachdruck der ersten Auflage von 1932, Grundlehren Math. Wiss. 38, Springer, Berlin, 1968.
• [8] G. K. Pedersen, Analysis Now, Grad. Texts in Math. 118, Springer, New York, 1989.
• [9] M. Reed and B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis, 2nd ed., Academic, New York, 1978.
• [10] K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Spaces, Grad. Texts in Math. 265, Springer, Dordrecht, 2012.
• [11] V. S. Sunder, Functional Analysis: Spectral Theory, Birkhäuser Adv. Texts Basler Lehrbücher, Birkhäuser, Basel, 1997.
• [12] S. L. Woronowicz, Unbounded elements affiliated with C∗-algebras and noncompact quantum groups, Commun. Math. Phys. 136 (1991), no. 2, 399–432.
• [13] S. L. Woronowicz and K. Napiórkowski, Operator theory in the C∗-algebra framework, Rep. Math. Phys. 31 (1992), no. 3, 353–371.