Displaying similar documents to “Brascamp--Lieb inequalities for non-commutative integration.”

Comparing quantum dynamical entropies

P. Tuyls (1998)

Banach Center Publications

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Last years, the search for a good theory of quantum dynamical entropy has been very much intensified. This is not only due to its usefulness in quantum probability but mainly because it is a very promising tool for the theory of quantum chaos. Nowadays, there are several constructions which try to fulfill this need, some of which are more mathematically inspired such as CNT (Connes, Narnhofer, Thirring), and the one proposed by Voiculescu, others are more inspired by physics such as...

Dynamical entropy of a non-commutative version of the phase doubling

Johan Andries, Mieke De Cock (1998)

Banach Center Publications

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A quantum dynamical system, mimicking the classical phase doubling map z ↦ z 2 on the unit circle, is formulated and its ergodic properties are studied. We prove that the quantum dynamical entropy equals the classical value log2 by using compact perturbations of the identity as operational partitions of unity.

Large time behaviour of a class of solutions of second order conservation laws

Jan Goncerzewicz, Danielle Hilhorst (2000)

Banach Center Publications

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% We study the large time behaviour of entropy solutions of the Cauchy problem for a possibly degenerate nonlinear diffusion equation with a nonlinear convection term. The initial function is assumed to have bounded total variation. We prove the convergence of the solution to the entropy solution of a Riemann problem for the corresponding first order conservation law.

Quantum dynamical entropy revisited

Thomas Hudetz (1998)

Banach Center Publications

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We define a new quantum dynamical entropy for a C*-algebra automorphism with an invariant state (and for an appropriate 'approximating' subalgebra), which entropy is a 'hybrid' of the two alternative definitions by Connes, Narnhofer and Thirring resp. by Alicki and Fannes (and earlier, Lindblad). We report on this entropy's properties and on three examples.

Non-commutative entropy computations for continuous fields and cross-products

Emmanuel Germain (2007)

Annales mathématiques Blaise Pascal

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We present here two non-commutative situations where dynamical entropy estimates are possible. The first result is concerned with automorphisms of cross-products by an exact group that commute with the group action and generalizes the result known for amenable groups. The second is about continuous fields of C * -algebras and C ( X ) -automorphisms. Each result relies on explicit factorization via matrices.

Quantum geometry of noncommutative Bernoulli shifts

Robert Alicki (1998)

Banach Center Publications

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We construct an example of a noncommutative dynamical system defined over a two dimensional noncommutative differential manifold with two positive Lyapunov exponents equal to ln d each. This dynamical system is isomorphic to the quantum Bernoulli shift on the half-chain with the quantum dynamical entropy equal to 2 ln d. This result can be interpreted as a noncommutative analog of the isomorphism between the classical one-sided Bernoulli shift and the expanding map of the circle and...