scholarly journals The free entropy dimension of hyperfinite von Neumann algebras

2003 ◽  
Vol 355 (12) ◽  
pp. 5053-5089 ◽  
Author(s):  
Kenley Jung
Keyword(s):  
Von Neumann Algebras ◽  
Von Neumann ◽  
Free Entropy ◽  
Entropy Dimension ◽  
Free Entropy Dimension

FREE ENTROPY DIMENSION OF PROJECTIONS AND FACTORIALITY OF GENERATED VON NEUMANN ALGEBRAS

2008 ◽  
Vol 11 (02) ◽  
pp. 295-305
Author(s):  
TAKUHO MIYAMOTO
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Sufficient Condition ◽  
Von Neumann ◽  
Free Entropy ◽  
Entropy Dimension ◽  
Free Entropy Dimension

We examine the free entropy and free entropy dimension for projections, and obtain a sufficient condition for the factoriality of the von Neumann algebra generated by projections in terms of their free entropy dimension. This corresponds to Voiculescu's result for self-adjoint elements.

scholarly journals Free entropy dimension and regularity of non-commutative polynomials

2016 ◽  
Vol 271 (8) ◽  
pp. 2274-2292 ◽  
Author(s):  
Ian Charlesworth ◽  
Dimitri Shlyakhtenko
Keyword(s):  
Free Entropy ◽  
Entropy Dimension ◽  
Free Entropy Dimension

scholarly journals A Random Matrix Approach to Absorption in Free Products

2020 ◽  
Author(s):  
Ben Hayes ◽  
David Jekel ◽  
Brent Nelson ◽  
Thomas Sinclair
Keyword(s):  
Random Matrix ◽  
Von Neumann Algebras ◽  
Concentration Of Measure ◽  
Unified Approach ◽  
Free Products ◽  
Von Neumann ◽  
Free Entropy ◽  
Famous Result ◽  
Random Matrix Models ◽  
Entropy Zero

Abstract This paper gives a free entropy theoretic perspective on amenable absorption results for free products of tracial von Neumann algebras. In particular, we give the 1st free entropy proof of Popa’s famous result that the generator MASA in a free group factor is maximal amenable, and we partially recover Houdayer’s results on amenable absorption and Gamma stability. Moreover, we give a unified approach to all these results using $1$-bounded entropy. We show that if ${\mathcal{M}} = {\mathcal{P}} * {\mathcal{Q}}$, then ${\mathcal{P}}$ absorbs any subalgebra of ${\mathcal{M}}$ that intersects it diffusely and that has $1$-bounded entropy zero (which includes amenable and property Gamma algebras as well as many others). In fact, for a subalgebra ${\mathcal{P}} \leq{\mathcal{M}}$ to have this absorption property, it suffices for ${\mathcal{M}}$ to admit random matrix models that have exponential concentration of measure and that “simulate” the conditional expectation onto ${\mathcal{P}}$.

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