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The degree one Laguerre-Pólya class and the shuffle-word-embedding conjecture
Journal article   Open access   Peer reviewed

The degree one Laguerre-Pólya class and the shuffle-word-embedding conjecture

James E. Pascoe and Hugo J. Woerdeman
Canadian mathematical bulletin
28 Feb 2024
url
https://doi.org/10.4153/S0008439524000146View
Published, Version of Record (VoR) Restricted

Abstract

Mathematics Physical Sciences Science & Technology
We discuss the class of functions, which are well approximated on compacta by the geometric mean of the eigenvalues of a unital (completely) positive map into a matrix algebra or more generally a type $II_1$ factor, using the notion of a Fuglede-Kadison determinant. In two variables, the two classes are the same, but in three or more noncommuting variables, there are generally functions arising from type $II_1$ von Neumann algebras, due to the recently established failure of the Connes embedding conjecture. The question of whether or not approximability holds for scalar inputs is shown to be equivalent to a restricted form of the Connes embedding conjecture, the so-called shuffle-word-embedding conjecture.

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