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Noncommutative independence in the infinite braid and symmetric group

100%

Gohm R. , Köstler C.

Banach Center Publications

2011

tom 96

nr 1

193-206

This is an introductory paper about our recent merge of a noncommutative de Finetti type result with representations of the infinite braid and symmetric group which allows us to derive factorization properties from symmetries. We explain some of the main ideas of this approach and work out a constructive procedure to use in applications. Finally we illustrate the method by applying it to the theory of group characters.

Stationary Quantum Markov processes as solutions of stochastic differential equations

80%

Hellmich J. , Köstler C. , Kümmerer B.

Banach Center Publications

1998

tom 43

nr 1

217-229

From the operator algebraic approach to stationary (quantum) Markov processes there has emerged an axiomatic definition of quantum white noise. The role of Brownian motion is played by an additive cocycle with respect to its time evolution. In this report we describe some recent work, showing that this general structure already allows a rich theory of stochastic integration and stochastic differential equations. In particular, if a quantum Markov process is represented by a unitary cocycle, we can reconstruct an additive cocycle ('quantum Brownian motion') and the unitary cocycle ('quantum Markov process') appears as the solution of a certain stochastic differential equation. This establishes a one-to-one correspondence between multiplicative and additive adapted cocycles. As an application of this result we construct stationary Markov processes, driven by squeezed white noise and q-white noise.