Lower estimates on microstates free entropy dimension
| dc.creator | Shlyakhtenko, D. | |
| dc.date | 2007-10-22 | |
| dc.date | 2008-07-02 | |
| dc.date.accessioned | 2026-07-07T09:47:55Z | |
| dc.date.available | 2026-07-07T09:47:55Z | |
| dc.description | By proving that certain free stochastic differential equations have stationary solutions, we give a lower estimate on the microstates free entropy dimension of certain $n$-tuples $X_{1},...,X_{n}$: we show that Abstract. By proving that certain free stochastic differential equations with analytic coefficients have stationary solutions, we give a lower estimate on the microstates free entropy dimension of certain n-tuples X_{1},...,X_{n}. In particular, we show that δ_{0}(X_{1},...,X_{n})\geq\dim_{M\bar{\otimes}M^{o}}V where M=W^{*}(X_{1},...,X_{n}) and V=\{(\partial(X_{1}),...,\partial(X_{n})):\partial\in\mathcal{C}\} is the set of values of derivations A=\mathbb{C}[X_{1},... X_{n}]\to A\otimes A with the property that \partial^{*}\partial(A)\subset A. We show that for q sufficiently small (depending on n) and X_{1},...,X_{n} a q-semicircular family, δ_{0}(X_{1},...,X_{n})>1. In particular, for small q, q-deformed free group factors have no Cartan subalgebras. An essential tool in our analysis is a free analog of an inequality between Wasserstein distance and Fisher information introduced by Otto and Villani (and also studied in the free case by Biane and Voiculescu). | |
| dc.description | A major revision. The previous version contained ad-hoc proofs for the case of q-semicircular variables. These have now been replaced with proofs based a more general approach involving free SDEs with analytic coefficients | |
| dc.identifier | https://arxiv.org/abs/0710.4111 | |
| dc.identifier | http://arxiv.org/abs/0710.4111 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/164023 | |
| dc.subject | Operator Algebras | |
| dc.subject | 46L54 | |
| dc.title | Lower estimates on microstates free entropy dimension | |
| dc.type | text |