scholarly journals Positive energy representations of Sobolev diffeomorphism groups of the circle

2020 ◽  
Vol 11 (1) ◽  
Author(s):  
Sebastiano Carpi ◽  
Simone Del Vecchio ◽  
Stefano Iovieno ◽  
Yoh Tanimoto
Keyword(s):  
Unitary Representation ◽  
Von Neumann Algebras ◽  
Universal Covering ◽  
Irreducible Representations ◽  
Positive Energy ◽  
Diffeomorphism Groups ◽  
Von Neumann ◽  
Direct Sum ◽  
Covering Groups

AbstractWe show that any positive energy projective unitary representation of $$\mathrm{Diff}_+(S^1)$$ Diff + ( S 1 ) extends to a strongly continuous projective unitary representation of the fractional Sobolev diffeomorphisms $$\mathcal {D}^s(S^1)$$ D s ( S 1 ) for any real $$s>3$$ s > 3 , and in particular to $$C^k$$ C k -diffeomorphisms $$\mathrm{Diff}_+^k(S^1)$$ Diff + k ( S 1 ) with $$k\ge 4$$ k ≥ 4 . A similar result holds for the universal covering groups provided that the representation is assumed to be a direct sum of irreducibles. As an application we show that a conformal net of von Neumann algebras on $$S^1$$ S 1 is covariant with respect to $$\mathcal {D}^s(S^1)$$ D s ( S 1 ) , $$s > 3$$ s > 3 . Moreover every direct sum of irreducible representations of a conformal net is also $$\mathcal {D}^s(S^1)$$ D s ( S 1 ) -covariant.

Lectures on von Neumann Algebras

2019 ◽  
Author(s):  
Serban-Valentin Stratila ◽  
Laszlo Zsido
Keyword(s):  
Von Neumann Algebras ◽  
Von Neumann

scholarly journals Approximate Tensorization of the Relative Entropy for Noncommuting Conditional Expectations

2021 ◽  
Author(s):  
Ivan Bardet ◽  
Ángela Capel ◽  
Cambyse Rouzé
Keyword(s):  
Relative Entropy ◽  
Von Neumann Algebras ◽  
Logarithmic Sobolev Inequality ◽  
Spin Systems ◽  
Von Neumann ◽  
Strong Subadditivity ◽  
Finite Dimensional ◽  
Lattice Spin ◽  
Lattice Spin Systems ◽  
Conditional Expectations

AbstractIn this paper, we derive a new generalisation of the strong subadditivity of the entropy to the setting of general conditional expectations onto arbitrary finite-dimensional von Neumann algebras. This generalisation, referred to as approximate tensorization of the relative entropy, consists in a lower bound for the sum of relative entropies between a given density and its respective projections onto two intersecting von Neumann algebras in terms of the relative entropy between the same density and its projection onto an algebra in the intersection, up to multiplicative and additive constants. In particular, our inequality reduces to the so-called quasi-factorization of the entropy for commuting algebras, which is a key step in modern proofs of the logarithmic Sobolev inequality for classical lattice spin systems. We also provide estimates on the constants in terms of conditions of clustering of correlations in the setting of quantum lattice spin systems. Along the way, we show the equivalence between conditional expectations arising from Petz recovery maps and those of general Davies semigroups.

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