Opuscula Math. 35, no. 2 (2015), 251-271
http://dx.doi.org/10.7494/OpMath.2015.35.2.251

Opuscula Mathematica

# Steering projections in von Neumann algebras

Abstract. A steering projection of an arbitrary von Neumann algebra is introduced. It is shown that a steering projection always exists and is unique (up to Murray-von Neumann equivalence). A general decomposition of arbitrary projections with respect to a steering projection is established.

Keywords: Murray-von Neumann order, central projection, steering projection.

Mathematics Subject Classification: 46L10.

Full text (pdf)

1. B. Blackadar, Operator Algebras. Theory of $$C^*$$-algebras and von Neumann Algebras, Springer-Verlag, Berlin, 2006.
2. E.L. Griffin Jr., Some contributions to the theory of rings of operators, Trans. Amer. Math. Soc. 75 (1953), 471-504.
3. E.L. Griffin Jr., Some contributions to the theory of rings of operators II, Trans. Amer. Math. Soc. 79 (1955), 389-400.
4. R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras, Vol. 1, Academic Press, New York, 1983.
5. R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras, Vol. 2, Academic Press, London, 1986.
6. P. Niemiec, Unitary equivalence and decompositions of finite systems of closed densely defined operators in Hilbert spaces, Dissertationes Math. (Rozprawy Mat.) 482 (2012).
7. S. Sakai, $$C^*$$-algebras and $$W^*$$-algebras, Springer-Verlag, Berlin, 1971.
8. J.T. Schwartz, $$W^*$$-algebras, Gordon and Breach, Science Publishers Inc., New York, 1967.
9. D. Sherman, On the dimension theory of von Neumann algebras, Math. Scand. 101 (2007), 123-147.
10. M. Takesaki, Theory of Operator Algebras I, Springer-Verlag, Berlin, 2002.
11. J. Tomiyama, Generalized dimension function for $$W^*$$-algebras of infinite type, Tôhoku Math. J. 10 (1958) 2, 121-129.
• Jagiellonian University, Faculty of Mathematics and Computer Science, ul. Łojasiewicza 6, 30-348 Krakow, Poland
• Communicated by Aurelian Gheondea.
• Revised: 2014-07-31.
• Accepted: 2014-08-06.
• Published online: 2014-11-18.