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On the CR-structure of certain linear group orbits in infinite dimensions

Wilhelm Kaup (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

For large classes of complex Banach spaces (mainly operator spaces) we consider orbits of finite rank elements under the group of linear isometries. These are (in general) real-analytic submanifolds of infinite dimension but of finite CR-codimension. We compute the polynomial convex hull of such orbits  M explicitly and show as main result that every continuous CR-function on  M has a unique extension to the polynomial convex hull which is holomorphic in a certain sense. This generalizes to infinite...

On the Lebesgue decomposition of the normal states of a JBW-algebra

Jacques Dubois, Brahim Hadjou (1992)

Mathematica Bohemica

In this article, a theorem is proved asserting that any linear functional defined on a JBW-algebra admits a Lebesque decomposition with respect to any normal state defined on the algebra. Then we show that the positivity (and the unicity) of this decomposition is insured for the trace states defined on the algebra. In fact, this property can be used to give a new characterization of the trace states amoungst all the normal states.

The structure and representation of n-ary algebras of DNA recombination

Sergei Sverchkov (2011)

Open Mathematics

In this paper we investigate the structure and representation of n-ary algebras arising from DNA recombination, where n is a number of DNA segments participating in recombination. Our methods involve a generalization of the Jordan formalization of observables in quantum mechanics in n-ary splicing algebras. It is proved that every identity satisfied by n-ary DNA recombination, with no restriction on the degree, is a consequence of n-ary commutativity and a single n-ary identity of the degree 3n-2....

Why Jordan algebras are natural in statistics: quadratic regression implies Wishart distributions

G. Letac, J. Wesołowski (2011)

Bulletin de la Société Mathématique de France

If the space 𝒬 of quadratic forms in ℝ n is splitted in a direct sum 𝒬 1 ⊕ ... ⊕ 𝒬 k and if X and Y are independent random variables of ℝ n , assume that there exist a real number a such that E ( X | X + Y ) = a ( X + Y ) and real distinct numbers b 1 , . . . , b k such that E ( q ( X ) | X + Y ) = b i q ( X + Y ) for any q in 𝒬 i . We prove that this happens only when k = 2 , when ℝ n can be structured in a Euclidean Jordan algebra and when X and Y have Wishart distributions corresponding to this structure.

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