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

Published electronically:
October 7, 2004

MathSciNet review:
2113934

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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]**Wai Mee Ching,*Free products of von Neumann algebras*, Trans. Amer. Math. Soc.**178**(1973), 147–163. MR**0326405**, 10.1090/S0002-9947-1973-0326405-3**[2]**D. W. Hadwin, W. E. Longstaff, and Peter Rosenthal,*Small transitive lattices*, Proc. Amer. Math. Soc.**87**(1983), no. 1, 121–124. MR**677246**, 10.1090/S0002-9939-1983-0677246-0**[3]**P. R. Halmos,*Ten problems in Hilbert space*, Bull. Amer. Math. Soc.**76**(1970), 887–933. MR**0270173**, 10.1090/S0002-9904-1970-12502-2**[4]**K. J. Harrison, Heydar Radjavi, and Peter Rosenthal,*A transitive medial subspace lattice*, Proc. Amer. Math. Soc.**28**(1971), 119–121. MR**0283609**, 10.1090/S0002-9939-1971-0283609-X**[5]**Richard V. Kadison and John R. Ringrose,*Fundamentals of the theory of operator algebras. Vol. I*, Pure and Applied Mathematics, vol. 100, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. Elementary theory. MR**719020**

Richard V. Kadison and John R. Ringrose,*Fundamentals of the theory of operator algebras. Vol. II*, Pure and Applied Mathematics, vol. 100, Academic Press, Inc., Orlando, FL, 1986. Advanced theory. MR**859186****[6]**M. S. Lambrou and W. E. Longstaff,*Small transitive families of subspaces in finite dimensions*, Linear Algebra Appl.**357**(2002), 229–245. MR**1935237**, 10.1016/S0024-3795(02)00380-4**[7]**D. V. Voiculescu, K. J. Dykema, and A. Nica,*Free random variables*, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, RI, 1992. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. MR**1217253**

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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.