SIGMA 2 (2006), 042, 19 pages      math.RT/0602677      https://doi.org/10.3842/SIGMA.2006.042

On Transitive Systems of Subspaces in a Hilbert Space

Yuliya P. Moskaleva ^a and Yurii S. Samoilenko ^b
a) Taurida National University, 4 Vernads'kyi Str., Simferopol, 95007 Ukraine
^b) Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka Str., Kyiv-4, 01601 Ukraine

Received February 27, 2006; Published online April 12, 2006

Abstract
Methods of *-representations in Hilbert space are applied to study of systems of n subspaces in a linear space. It is proved that the problem of description of n-transitive subspaces in a finite-dimensional linear space is *-wild for n ≥ 5.

Key words: algebras generated by projections; irreducible inequivalent representations; transitive nonisomorphic systems of subspaces.

pdf (301 kb)   ps (219 kb)   tex (18 kb)

References

1. Halmos P.R., Two subspaces, Trans. Amer. Math. Soc., 1969, V.144, 381-389.
2. Halmos P.R., Ten problems in Hilbert space, Bull. Amer. Math. Soc., 1970, V.76, 887-933.
3. Brenner S., Endomorphism algebras of vector spaces with distinguished sets of subspaces, J. Algebra, 1967, V.6, 100-114.
4. Gel'fand I.M., Ponomarev V.A., Problems of linear algebra and classification of quadruples of subspaces in a finite-dimensional vector space, Coll. Math. Spc. Bolyai 5, Tihany, 1970, 163-237.
5. Nazarova L.A., Representations of a quadruple, Izv. AN SSSR, 1967, V.31, N 6, 1361-1377 (in Russian).
6. Enomoto M., Watatani Ya., Relative position of four subspaces in a Hilbert space, math.OA/0404545.
7. Kruglyak S.A., Rabanovich V.I., Samolenko Yu.S., On sums of projections, Funktsional. Anal. i Prilozhen., 2002, V.36, N 3, 30-35 (English transl.: Funct. Anal. Appl., 2002, V.36, N 3, 182-195).
8. Kruglyak S.A., Samolenko Yu.S., On the complexity of description of representations of *-algebras generated by idempotents, Proc. Amer. Math. Soc., 2000, V.128, 1655-1664.
9. Ostrovskyi V.L., Samolenko Yu.S., Introduction to the theory of representations of finitely presented *-algebras. I. Representations by bounded operators, Harwood Acad. Publs., 1999.
10. Popovich S.V., Samolenko Yu.S., On homomorphisms of algebras generated by projections, and the Coxeter functors, Ukrain. Mat. Zh., 2003, V.55, N 9, 1224-1237 (English transl.: Ukrainian Math. J., 2003, V.55, N 9, 1480-1496).
11. Moskaleva Yu.P., Samolenko Yu.S., Systems of n subspaces and representations of *-algebras generated by projections, Methods Funct. Anal. Topology, 2006, V.12, N 1, 57-73.
12. Moskaleva Yu.P., On *-representations of the algebra P_4,abo,t, Uchenye Zapiski Tavricheskogo Natsional'nogo Universiteta imeni Vernadskogo, Seriya Matem. Mech. Inform. Kibern., 2005, N 1, 27-35 (in Russian).