00-459 Madalin Guta, Hans Maassen

Generalised Brownian motion and second quantisation (126K, latex 2e with 5 figures) Nov 20, 00

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Abstract. A new approach to the generalised Brownian motion introduced by M. Bo\.zejko and R. Speicher is described, based on symmetry rather than deformation. The symmetrisation principle is provided by Joyal's notions of tensorial and combinatorial species. Any such species V gives rise to an endofunctor F_V of the category of Hilbert spaces with contractions mapping a Hilbert space H to a symmetric Hilbert space F_V(H) with the same symmetry as the species V. A general framework for annihilation and creation operators on these spaces is developed and shown to give vacuum expectations as prescribed by Bo\.zejko and Speicher. The existence of the second quantisation as functor from Hilbert spaces to von Neumann algebras with completely positive maps is investigated. For a certain one parameter interpolation between the classical and the free Brownian motion it is shown that the ``field algebras'' Gamma(K) are type II_1 factors when K is infinite dimensional.

Files: 00-459.src( 00-459.keywords , gbm.tex , adjoint.eps , decomposition.eps , diagram.eps , permutinglegs.eps , product.eps )