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Schwarz inequalities and the decomposition of positive maps on C*-algebras

Published online by Cambridge University Press:  24 October 2008

A. Guyan Robertson
A. Guyan Robertson
Affiliation:
Department of Mathematics, University of Edinburgh, Edinburgh EH9 3JZ
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Extract

In recent years there has been considerable progress in the study of certain linear maps of C*-algebras which preserve the natural partial ordering. The most tractable such maps, the completely positive ones, have proved to be of great importance in the structure theory of C*-algebras(4). However general positive (order-preserving) linear maps are (at present) very intractable. For example, there is no algebraic formula which enables one to construct a general positive map, even on the algebra of 3 3 complex matrices. It is therefore of interest to study conditions stronger than positivity, but weaker than complete positivity.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 94 , Issue 2 , September 1983 , pp. 291 - 296
Copyright
Copyright © Cambridge Philosophical Society 1983

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References

REFERENCES

(1)Choi, M. D.Positive linear maps on C*-algebras. Canad. J. Math. 24 (1972), 620529.CrossRefGoogle Scholar
(2)Choi, M. D.A Schwarz inequality for positive linear maps on C*-algebras. Illinois J. Math. 18 (1974), 565574.CrossRefGoogle Scholar
(3)Choi, M. D.Some assorted inequalities for positive linear maps on C*-algebras. J. Operator Theory 4 (1980), 271285.Google Scholar
(4)Choi, M. D. and Effros, E. G.Injectivity and operator spaces. J. Functional Analysis 24 (1977), 156209.CrossRefGoogle Scholar
(5)Choi, M. D. and Lam, T. Y.Extremal positive semidefinite forms. Math. Ann. 231 (1977), 118.CrossRefGoogle Scholar
(6)Hamana, M.Injective envelopes of C*-algebras. J. Math. Soc. Japan 31 (1979), 181197.Google Scholar
(7)Kirchberg, E. The strong Kadison inequality of Woronowicz. Akademie der Wissen-schaften der DDR, preprint P 1178.Google Scholar
(8)O'Brien, R. C.On extreme matrices and extreme vectors of cones in Rn. Linear Algebra Appl. 12 (1975), 7779.CrossRefGoogle Scholar
(9)Robertson, A. G.Automorphisms of spin factors and the decomposition of positive maps. Quart. J. Math. 34 (1983), 8796.CrossRefGoogle Scholar
(10)Robertson, A. G.Positive extensions of automorphisms of spin factors. Proc. Roy. Soc. Edinburgh Sect. A 94 (1983), 7177.Google Scholar
(11)Stormer, E.Positive linear maps of operator algebras. Acta Math. 110 (1963), 233278.CrossRefGoogle Scholar
(12)Stormer, E.Decomposable positive maps on C*-algebras. Proc. Amer. Math. Soc. 86 (1982), 402404.Google Scholar
(13)Woronowicz, S. L.Positive maps of low dimensional matrix algebras. Rep. Math. Phys. 10 (1976), 165183.CrossRefGoogle Scholar
(14)Woronowicz, S. L.Nonextendible positive maps. Comm. Math. Phys. 51 (1976), 243282.CrossRefGoogle Scholar