Geodesic
An Inequality Characterizes the Trace
Canadian journal of mathematics, Tome 31 (1979) no. 6, pp. 1322-1328

Voir la notice de l'article provenant de la source Cambridge University Press

1. Introduction. While analogues of the Schwarz inequality have been much studied in the context of positive linear maps of operator algebras ([1], [2], [6], [7], [10]) the simpler triangle inequality |φ(x)| ≦ (|x|) has been neglected, outside of (possibly non-commutative) integration theory—perhaps partly because except for the important and familiar example of traces, scalar maps satisfying the triangle inequality are rarely encountered. In fact we here prove that they are never encountered: every such map is a trace.For C*-algebras (norm-closed self-ad joint algebras of bounded operators on a Hilbert space) this means, for instance, that if the linear functional φ on the C*-algebra satisfies (†) then φ satisfies also the equivalent conditions (i) φ(xy) = φ(yx) for all x, y in ; (ii) φ(x*x) = φ(xx*) for all x in ; (iii) φ(x) = φ(uxu*) for all x in and all unitary u in Ae, the C*-lgebra formed from by adjunction of a unit element. 
Gardner, L. Terrell. An Inequality Characterizes the Trace. Canadian journal of mathematics, Tome 31 (1979) no. 6, pp. 1322-1328. doi: 10.4153/CJM-1979-109-9
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