Wednesday , December 19 2018

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.

>>Full text
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 Image984-2009,1,8. 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 Image985-2009,1,8(abcd), a,b,c,d Image986-2009,1,8V) into with the normalized trace.

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

REFERENCES

  1. [Co] Connes, A., Classification of injective factors. Cases II, II, IIIλ , λ=1, Ann. of Math. (2) 104 (1976), No. 1, pp. 73-115.
  2. [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.
  3. [Vo] Voiculescu, D., Circular and semicircular systems and free product factors, in Progress in Math., Vol. 92, Birkhäuser, 1990.