Cyclic Hilbert Spaces

Florin RĂDULESCU
Universita Roma “Tor Vergata”
Institute of Mathematics, Romanian Academy

Dedicated to Professor Andrei Neculai to his 60th birthday

Abstract: We analyse in this paper a concept related to the Connes Embedding Problem [Co]. A type II algebra is an algebra with a trace, and CEP requires for the multiplication to be approximated by matrices. Here we start the analysis of four products, which is the study of cyclic Hilbert spaces.

Keywords:

Cyclic Hilbert space, connes embedding problem.

Florin Rădulescu Born 15.08.1960 in Bucharest. Studies University of Bucharest, PhD in Mathematics Univ. of California at Los Angeles 1991. Positions held : Full Professor Univ of Iowa (1996- 2008, associate 1994-1996), Full Professor Univ of Rome Tor Vergata since 2002. Member of Institute of Mathematics Romanian Academy since 1985 (CP1 since 2002). 5 PhD students at the Univ of Iowa that graduated before 2005. Presently supervising two Ph.D students at Uni. Rome. Principal investigator for three consecutive three years NSF grants, director of a CEEX grant 2006-2008. Price Simion Stoilow of the Romanian Academy for the paper “Fundamental group of the von Neuman algebra of a free group with infinitely many generators is R_+{0}”. 33 papers published, the most cited being “Random Matrices, Amalgamated Free products and subfactors published In Inventiones Matematicae. Interest: Operator Algebras in connection with Number Theory.

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CITE THIS PAPER AS:
Florin RĂDULESCU, Cyclic Hilbert Spaces, Studies in Informatics and Control, ISSN 1220-1766, vol. 18 (1), pp. 83-86, 2009.

In this paper we introduce the notion of a cyclic Hilbert space, which is by definition a Hilbert space, that carries a special cyclic scalar product on . We prove that such spaces can be embedded into finite unbounded (separable) von Neumann algebras.

Given are arbitrary II factor M, and V a subspace of selfadjoint elements, the Connes embedding Problem is reducible ([Ra]) to the problem to approximation of four products: that is if V is a finite dimensional real vector space of M, find an approximate embedding (that preserves approximately (abcd), a,b,c,d V) into with the normalized trace.

This consists into proving that every cyclic Hilbert Space, as defined bellow is embeddable into a II factor.

REFERENCES

[Co] Connes, A., Classification of injective factors. Cases II, II_∞, III_λ, λ=1, Ann. of Math. (2) 104 (1976), No. 1, pp. 73-115.
[Ra] Rădulescu, F., A non-commutative, analytic version of Hilbert’s 17th problem in type II von Neumann algebras, math.OA/0404458, To appear in Proceedings Theta Foundation.
[Vo] Voiculescu, D., Circular and semicircular systems and free product factors, in Progress in Math., Vol. 92, Birkhäuser, 1990.