Journal of Convex Analysis 29 (2022), No. 3, 755--766
Copyright Heldermann Verlag 2022
Proscribed Normal Decompositions of Euclidean Jordan Algebras
Dept. of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, U.S.A.
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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