Michael Skeide

ON THE RELATION OF THE SQUARE OF WHITE NOISE AND THE FINITE DIFFERENCE ALGEBRA

The algebra of square of white noise1 contains a subalgebra generated by elements fulfilling the relations of Feinsilvers finite difference algebra.6 Moreover, Boukas representation space3 is the same as the representation space of the algebra of square of white noise discovered in Ref. 2. In other words, Boukas representation extends to a representation of the algebra of square of white noise.

Infinitesimal Generators on the Quantum Group SUq(2)

Quantum Levy processes on a quantum group are, like classical Levy processes with values in a Lie group, classified by their infinitesimal generators. We derive a formula for the infinitesimal generators on the quantum group SUq(2) and decompose them in terms of an infinite-dimensional irreducible representation and of characters. Thus we obtain a quantum Levy–Khintchine formula.

CONSTRUCTING EXTENSIONS OF CP-MAPS VIA TENSOR DILATIONS WITH THE HELP OF VON NEUMANN MODULES

We apply Hilbert module methods to show that normal completely positive maps admit weak tensor dilations. Appealing to a duality between weak tensor dilations and extensions of CP-maps, we get an existence proof for certain extensions. We point out that this duality is part of a far reaching duality between a von Neumann bimodule and its commutant in which other dualities, known and new, also find their natural common place.

A CENTRAL LIMIT THEOREM FOR BOSE

We present a central limit theorem for Bose -independent operator-valued random variables. Furthermore, we show that the central limit distribution may be represented by an algebra of creators and annihilators on a symmetric Fock module. As an example we recover the distribution of creators and annihilators on the truncated Fock space, i.e. the central limit distribution of Boolean independence.