Remarks on free entropy dimension
Abstract
We prove a technical result, showing that the existence of a closable unbounded dual system in the sense of Voiculescu is equivalent to the finiteness of free Fisher information. This approach allows one to give a purely operator-algebraic proof of the computation of the non-microstates free entropy dimension for generators of groups carried out in an earlier joint work with I. Mineyev. The same technique also works for finite-dimensional algebras. We also show that Voiculescu's question of semi-continuity of free entropy dimension, as stated, admits a counterexample. We state a modified version of the question, which avoids the counterexample, but answering which in the affirmative would still imply the non-isomorphism of free group factors.
Keywords
Cite
@article{arxiv.math/0504062,
title = {Remarks on free entropy dimension},
author = {Dimitri Shlyakhtenko},
journal= {arXiv preprint arXiv:math/0504062},
year = {2007}
}