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arXiv · 1402.6948

Logarithmic Sobolev inequalities and exponential entropy decay in non-commutative algebras

Raffaella Carbone
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Abstract

We study the relations between (tight) logarithmic Sobolev inequalities, entropy decay and spectral gap inequalities for Markov evolutions on von Neumann algebras. We prove that log-Sobolev inequalities (in the non-commutative form defined by Olkiewicz and Zegarlinski) imply spectral gap inequalities, with optimal relation between the constants. Furthermore, we show that a uniform exponential decay of a proper relative entropy is equivalent to a modified version of log-Sobolev inequalities; this entropy decay turns out to be implied by the usual log-Sobolev inequality adding some regularity conditions on the quadratic forms.

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Raffaella Carbone. 2014-02-27. Logarithmic Sobolev inequalities and exponential entropy decay in non-commutative algebras. https://arxiv.org/abs/1402.6948

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