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Relations and trails in lattices of projections in W*-algebras

Part of: Selfadjoint operator algebras Modular lattices, complemented lattices

Published online by Cambridge University Press:  23 August 2019

Edward Kissin
Edward Kissin*
Affiliation:
STORM, School of Computing and Digital Media, London Metropolitan University, 166-220 Holloway Road, London N7 8DB, UK (e.kissin@londonmet.ac.uk)
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Abstract

This paper concerns HH-relations in the lattices P(M) of all projections of W*-algebras M. If M is a finite algebra, all these relations are generated by trails in P(M). If M is an infinite countably decomposable factor, they are either generated by trails or associated with them.

Keywords

latticeHH-relation on a latticeradicalW*-algebraprojectiontrailtrace

MSC classification

Primary: 46L10: General theory of von Neumann algebras
Secondary: 06C99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 4 , November 2019 , pp. 1189 - 1215
Copyright
Copyright © Edinburgh Mathematical Society 2019 

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References

1.Amitsur, S. A., A general theory of radicals. I. Radicals in complete lattices, Amer. J. Math. 74 (1952), 774–786.Google Scholar
2.Amitsur, S. A., A general theory of radicals. II. Radicals in rings and bicategories, Amer. J. Math. 76 (1954), 100–125.Google Scholar
3.Amitsur, S. A., A general theory of radicals. III. Applications, Amer. J. Math. 76 (1954), 126–136.Google Scholar
4.Dixon, P. G., Topologically irreducible representations and radicals in Banach algebras, Proc. Lond. Math. Soc. (3) 74 (1997), 174–200.Google Scholar
5.Gray, M., A radical approach to algebra (Addison-Wesley, London–Ontario, 1970).Google Scholar
6.Kadison, R. V. and Ringrose, J. R., Fundamentals of the theory of operator algebras Vol. I and II (Academic Press, London, 1986).Google Scholar
7.Kissin, E., Shulman, V. S. and Turovskii, Yu. V., Topological radicals and Frattini theory of Banach Lie algebras, Integral Equations and Operator Theory 77 (2012), 51–121.Google Scholar
8.Kissin, E., Shulman, V. S. and Turovskii, Yu. V., On theory of topological radicals, Contemp. Math., Fund. Dir. 64 (3)(2018), 490–546.Google Scholar
9.Kissin, E., Shulman, V. S. and Turovskii, Yu. V., Relations and radicals in abstract lattices and in lattices of subspaces of Banach spaces and ideals of Banach algebras. Amitsur's theory revisited (arXiv: 1903.12291, 2019).Google Scholar
10.Kissin, E., Shulman, V. S. and Turovskii, Yu. V., Relations and radicals in lattices of ideals of C*-algebras (in press).Google Scholar
11.Kurosh, A. G., Radicals of rings and of algebras, Mat. Sb. 33 (1953), 13–26.Google Scholar
12.Naimark, M. A., Normed algebras (Wolters-Noordhoff Publishing, Netherlands, 1972).Google Scholar
13.Sakai, S., C*-algebras and W*-algebras (Springer-Verlag, New York–Heidelberg, 1998).Google Scholar
14.Shulman, V. S. and Turovskii, Yu. V., Topological radicals, V. From algebra to spectral theory, operator theory advances and applications, Algebr. Methods Funct. Anal. 233 (2014), 171–280.Google Scholar
15.Takesaki, M., Theory of operator algebras I (Springer-Verlag, New York–Heidelberg, 2002).Google Scholar