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An Inequality Characterizes the Trace

Published online by Cambridge University Press:  20 November 2018

L. Terrell Gardner
L. Terrell Gardner*
Affiliation:
University of Toronto, Toronto, Ontario
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 6 , 01 December 1979 , pp. 1322 - 1328
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Choi, M.-D., A Schwarz inequality for positive linear maps on C*-algebras, Illinois J. Math. 18 (1974), 565–574.Google Scholar
2. Davis, Ch., A Schwarz inequality for convex operator functions, Proc. Amer. Math. Soc. 8 (1957), 42–44.Google Scholar
3. Dixmier, J., Les algèbres d'operateurs dans Vespace Hilbertien (algèbres de von Neumann), Paris, 1957.Google Scholar
4. Dixmier, J., Les C*-algèbres et leurs représentations, Paris, 1964.Google Scholar
5. Gardner, L. T., A dilatation theorem for |• |-preserving maps of C*'-algebras, Proc. Amer. Math. Soc. 73 (1979), 461–465.Google Scholar
6. Gardner, L. T., Linear maps of C*-algebras preserving the absolute value, Proc. Amer. Math. Soc. 76 (1979), 271–278.Google Scholar
7. Kadison, R. V., A generalized Schwarz inequality and algebraic invariants for operator algebras, Annals of Math. 56 (1952), 494–503.Google Scholar
8. Kaplansky, I., A theorem on rings of operators, Pacific J. Math. 1 (1951), 227–232.Google Scholar
9. Sherman, S., The second adjoint of a C*-algebra, Proc. Int. Congr. Math., Cambridge 1 (1950), 470.Google Scholar
10. Stormer, E., Positive linear maps of C*-algebras, Lecture Notes in Physics 29 (1974), 85–106.Google Scholar