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Register a new userConformal Field Theory and Operator Algebras
تحليل الحقل المطابق وألحان المشغل
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Added by
Yasuyuki Kawahigashi
Publication date
2007
fields
Physics
and research's language is
English
Authors
Yasuyuki Kawahigashi
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نحن نستعرض التقدم الحديث في النهج الخطي العامل لنظرية المجال الكونفورمي الكمي. نحن نركز على استخدام نظرية التمثيل في نظرية التصنيف. وهذا يستند إلى سلسلة من الأعمال المشتركة مع ر. لونغو.
We review recent progress in operator algebraic approach to conformal quantum field theory. Our emphasis is on use of representation theory in classification theory. This is based on a series of joint works with R. Longo.
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As a continuation of the paper [20] on standard $f$-divergences, we make a systematic study of maximal $f$-divergences in general von Neumann algebras. For maximal $f$-divergences, apart from their definition based on Haagerups $L^1$-space, we present the general integral expression and the variational expression in terms of reverse tests. From these definition and expressions we prove important properties of maximal $f$-divergences, for instance, the monotonicity inequality, the joint convexity, the lower semicontinuity, and the martingale convergence. The inequality between the standard and the maximal $f$-divergences is also given.
We make a systematic study of standard $f$-divergences in general von Neumann algebras. An important ingredient of our study is to extend Kosakis variational expression of the relative entropy to an arbitary standard $f$-divergence, from which most of the important properties of standard $f$-divergences follow immediately. In a similar manner we give a comprehensive exposition on the Renyi divergence in von Neumann algebra. Some results on relative hamiltonians formerly studied by Araki and Donald are improved as a by-product.
A lemma stated by Ke Li in [arXiv:1208.1400] has been used in e.g. [arXiv:1510.04682,arXiv:1706.04590,arXiv:1612.01464,arXiv:1308.6503,arXiv:1602.08898] for various tasks in quantum hypothesis testing, data compression with quantum side information or quantum key distribution. This lemma was originally proven in finite dimension, with a direct extension to type I von Neumann algebras. Here we show that the use of modular theory allows to give more transparent meaning to the objects constructed by the lemma, and to prove it for general von Neumann algebras. This yields immediate generalizations of e.g. [arXiv:1510.04682].
We extend the new perturbation formula of equilibrium states by Hastings to KMS states of general $W^*$-dynamical systems.
Continuous groups with antilinear operations of the form $G+a_0G$, where $G$ denotes a linear Lie group, and $a_0$ is an antilinear operation which fulfills the condition $a^2_0=pm 1$, were defined and their matrix algebras were investigated in cite{Kocinski4}. In this paper infinitesimal-operator algebras are defined for any group of the form $G+a_0G$, and their properties are determined.
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