Journal of Convex Analysis 29 (2022), No. 3, 755--766
Copyright Heldermann Verlag 2022
Proscribed Normal Decompositions of Euclidean Jordan Algebras
Michael Orlitzky
Dept. of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, U.S.A.
michael@orlitzky.com
Normal decomposition systems unify many results from convex matrix analysis regarding functions that are invariant with respect to a group of transformations -- particularly those matrix functions that are unitarily-invariant and the affiliated permutation-invariant "spectral functions" that depend only on eigenvalues. Spectral functions extend in a natural way to Euclidean Jordan algebras, and several authors have studied the problem of making a Euclidean Jordan algebra into a normal decomposition system. In particular it is known to be possible with respect to the "eigenvalues of" map when the algebra is essentially-simple. We show the converse, that essential-simplicity is essential to that process.
Keywords: Normal decomposition system, Eaton triple, spectral function, group majorization, Euclidean Jordan algebra.
MSC: 17C20, 17C30, 17C55, 52A41, 90C25.
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