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The von Neumann algebras generated by t -gaussians

Éric Ricard — 2006

Annales de l’institut Fourier

We study the t -deformation of gaussian von Neumann algebras. They appear as example in the theories of Interacting Fock spaces and conditionally free products. When the number of generators is fixed, it is proved that if t is sufficiently close to 1 , then these algebras do not depend on t . In the same way, the notion of conditionally free von Neumann algebras often coincides with freeness.

On the algebraic structure of the unitary group.

Éric RicardChristian Rosendal — 2007

Collectanea Mathematica

We consider the unitary group U of complex, separable, infinite-dimensional Hilbert space as a discrete group. It is proved that, whenever U acts by isometries on a metric space, every orbit is bounded. Equivalently, U is not the union of a countable chain of proper subgroups, and whenever E ⊆ U generates U, it does so by words of a fixed finite length.

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