Please use this identifier to cite or link to this item: doi:10.22028/D291-30685
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Title: Absence of algebraic relations and of zero divisors under the assumption of full non-microstates free entropy dimension
Author(s): Mai, Tobias
Speicher, Roland
Weber, Moritz
Language: English
Title: Advances in mathematics
Volume: 304
Startpage: 1080
Endpage: 1107
Publisher/Platform: Elsevier
Year of Publication: 2017
Publikation type: Journal Article
Abstract: We show that in a tracial and finitely generated W⁎ -probability space existence of conjugate variables excludes algebraic relations for the generators. Moreover, under the assumption of maximal non-microstates free entropy dimension, we prove that there are no zero divisors in the sense that the product of any non-commutative polynomial in the generators with any element from the von Neumann algebra is zero if and only if at least one of those factors is zero. In particular, this shows that in this case the distribution of any non-constant self-adjoint non-commutative polynomial in the generators does not have atoms. Questions on the absence of atoms for polynomials in non-commuting random variables (or for polynomials in random matrices) have been an open problem for quite a while. We solve this general problem by showing that maximality of free entropy dimension excludes atoms.
DOI of the first publication: 10.1016/j.aim.2016.09.018
URL of the first publication: https://www.sciencedirect.com/science/article/pii/S0001870815301183
Link to this record: hdl:20.500.11880/28963
http://dx.doi.org/10.22028/D291-30685
ISSN: 1090-2082
0001-8708
Date of registration: 6-Apr-2020
Faculty: MI - Fakultät für Mathematik und Informatik
Department: MI - Mathematik
Professorship: MI - Prof. Dr. Roland Speicher
Collections:UniBib – Die Universitätsbibliographie

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