scholarly journals Uniform mixing and completely positive sofic entropy

2019 ◽  
Vol 138 (2) ◽  
pp. 597-612
Author(s):  
Tim Austin ◽  
Peter Burton
Keyword(s):  
Completely Positive ◽  
Sofic Entropy

scholarly journals The Koopman Representation and Positive Rokhlin Entropy

2021 ◽  
Author(s):  
Brandon Seward
Keyword(s):  
Spectral Gap ◽  
Regular Representation ◽  
Positive Entropy ◽  
Completely Positive ◽  
Amenable Groups ◽  
Direct Sum ◽  
Left Regular Representation ◽  
Left Regular ◽  
Sofic Entropy ◽  
Final Consequence

Abstract In this paper, we study connections between positive entropy phenomena and the Koopman representation for actions of general countable groups. Following the line of work initiated by Hayes for sofic entropy, we show in a certain precise manner that all positive entropy must come from portions of the Koopman representation that embed into the left-regular representation. We conclude that for actions having completely positive outer entropy, the Koopman representation must be isomorphic to the countable direct sum of the left-regular representation. This generalizes a theorem of Dooley–Golodets for countable amenable groups. As a final consequence, we observe that actions with completely positive outer entropy must be mixing, and when the group is non-amenable they must be strongly ergodic and have spectral gap.

scholarly journals POLISH MODELS AND SOFIC ENTROPY

2016 ◽  
Vol 17 (2) ◽  
pp. 241-275 ◽  
Author(s):  
Ben Hayes
Keyword(s):  
Probability Measure ◽  
Spectral Gap ◽  
A Priori ◽  
Positive Entropy ◽  
Completely Positive ◽  
Orbit Equivalent ◽  
Factor Theorem ◽  
Sofic Entropy ◽  
Measure Preserving Actions ◽  
Theoretic Version

We deduce properties of the Koopman representation of a positive entropy probability measure-preserving action of a countable, discrete, sofic group. Our main result may be regarded as a ‘representation-theoretic’ version of Sinaǐ’s factor theorem. We show that probability measure-preserving actions with completely positive entropy of an infinite sofic group must be mixing and, if the group is nonamenable, have spectral gap. This implies that if$\unicode[STIX]{x1D6E4}$is a nonamenable group and$\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$is a probability measure-preserving action which is not strongly ergodic, then no action orbit equivalent to$\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$has completely positive entropy. Crucial to these results is a formula for entropy in the presence of a Polish, but a priori noncompact, model.

On the cp-Rank and Minimal cp Factorizations of a Completely Positive Matrix

2013 ◽  
Vol 34 (2) ◽  
pp. 355-368 ◽  
Author(s):  
Naomi Shaked-Monderer ◽  
Immanuel M. Bomze ◽  
Florian Jarre ◽  
Werner Schachinger
Keyword(s):  
Positive Matrix ◽  
Completely Positive ◽  
Completely Positive Matrix

BEYOND STRONG SUBADDITIVITY? IMPROVED BOUNDS ON THE CONTRACTION OF GENERALIZED RELATIVE ENTROPY

1994 ◽  
Vol 06 (05a) ◽  
pp. 1147-1161 ◽  
Author(s):  
MARY BETH RUSKAI
Keyword(s):  
Relative Entropy ◽  
Discrete Case ◽  
Sobolev Inequalities ◽  
Completely Positive ◽  
Logarithmic Sobolev Inequalities ◽  
Strong Subadditivity

New bounds are given on the contraction of certain generalized forms of the relative entropy of two positive semi-definite operators under completely positive mappings. In addition, several conjectures are presented, one of which would give a strengthening of strong subadditivity. As an application of these bounds in the classical discrete case, a new proof of 2-point logarithmic Sobolev inequalities is presented in an Appendix.

Positive Linear Maps on C*-Algebras

1972 ◽  
Vol 24 (3) ◽  
pp. 520-529 ◽  
Author(s):  
Man-Duen Choi
Keyword(s):  
Vector Spaces ◽  
Complex Numbers ◽  
Complex Matrices ◽  
Completely Positive ◽  
C Algebras ◽  
Linear Maps ◽  
Positive Linear Maps

The objective of this paper is to give some concrete distinctions between positive linear maps and completely positive linear maps on C*-algebras of operators.Herein, C*-algebras possess an identity and are written in German type . Capital letters A, B, C stand for operators, script letters for vector spaces, small letters x, y, z for vectors. Capital Greek letters Φ, Ψ stand for linear maps on C*-algebras, small Greek letters α, β, γ for complex numbers.We denote by the collection of all n × n complex matrices. () = ⊗ is the C*-algebra of n × n matrices over .

scholarly journals Completely positive Bloch-Boltzmann equations

2003 ◽  
Vol 68 (1) ◽  
Author(s):  
Robert Alicki ◽  
Stanisław Kryszewski
Keyword(s):  
Boltzmann Equations ◽  
Completely Positive
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