Transitive families of projections in factors of type

Author:
Jon P. Bannon

Journal:
Proc. Amer. Math. Soc. **133** (2005), 835-840

MSC (2000):
Primary 46L54; Secondary 47A62

DOI:
https://doi.org/10.1090/S0002-9939-04-07717-2

Published electronically:
October 7, 2004

MathSciNet review:
2113934

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce a notion of transitive family of subspaces relative to a type factor, and hence a notion of transitive family of projections in such a factor. We show that whenever is a factor of type and is generated by two self-adjoint elements, then contains a transitive family of projections. Finally, we exhibit a free transitive family of projections that generate a factor of type .

**[1]**W.M. Ching, ``Free products of von Neumann algebras", Trans. Amer. Math. Soc. 178 (1973), 147-163. MR**0326405 (48:4749)****[2]**D.W. Hadwin, W.E. Longstaff and Peter Rosenthal, ``Small transitive lattices", Proc. Amer. Math. Soc. 87 (1983), 121-124. MR**0677246 (85e:47002)****[3]**P.R.Halmos, ``Ten Problems in Hilbert Space", Bull. Amer. Math. Soc. 76 (1970) 887-933. MR**0270173 (42:5066)****[4]**K.J. Harrison, Heydar Radjavi and Peter Rosenthal, ``A Transitive Medial Subspace Lattice", Proc. Amer. Math. Soc. 28 (1971), 119-121. MR**0283609 (44:839)****[5]**R.Kadison and J. Ringrose, ``Fundamentals of the Theory of Operator Algebras", vols. I and II, Academic Press, Orlando, FL, 1983 and 1986. MR**0719020 (85j:46099)**; MR**0859186 (88d:46106)****[6]**M.S. Lambrou, W.E. Longstaff, ``Small transitive families of subspaces in finite dimensions", Lin. Alg. Appl. 357 (2002), 229-245. MR**1935237 (2003m:47012)****[7]**Voiculescu, D.V.; Dykema K.J.; Nica, A., ``Free random variables. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups.", CRM monograph series, I. American Mathematical Society, Providence, RI, 1992. MR**1217253 (94c:46133)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
46L54,
47A62

Retrieve articles in all journals with MSC (2000): 46L54, 47A62

Additional Information

**Jon P. Bannon**

Affiliation:
Department of Mathematics and Statistics, The University of New Hampshire, Dur- ham, New Hampshire 03872

Email:
jpbannon@math.unh.edu

DOI:
https://doi.org/10.1090/S0002-9939-04-07717-2

Keywords:
II$_{1}$ factor,
transitive family,
free product

Received by editor(s):
November 18, 2003

Published electronically:
October 7, 2004

Communicated by:
David R. Larson

Article copyright:
© Copyright 2004
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.