**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, **C****yclic 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.