Geodesic
Dilations and Hahn Decompositions for Linear Maps
Canadian journal of mathematics, Tome 33 (1981) no. 4, pp. 826-839

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

Suppose is a C*-algebra and H is a Hilbert space. Let denote the set of completely positive maps from into the set B(H) of (bounded linear) operators on H. This paper studies the vector space spanned by , i.e., the linear maps that are finite linear combinations of completely positive maps. From another viewpoint, a map φ is in precisely when it has a decomposition φ = (φ1 – φ2) + i(φ3 – φ4) with φ1, φ2, φ3, φ4 in CP ; this decomposition is analogous to the Hahn decomposition for measures [8, 111.4.10] (see also Theorem 20). The analogous class of maps with “completely positive” replaced by “positive” was studied by R. I. Loebl [11] and S.-K. Tsui [17], and when is commutative, this latter class coincides withi , since every positive linear map on a commutative C*-algebra is completely positive [16]. 
Hadwin, D. W. Dilations and Hahn Decompositions for Linear Maps. Canadian journal of mathematics, Tome 33 (1981) no. 4, pp. 826-839. doi: 10.4153/CJM-1981-064-7
@article{10_4153_CJM_1981_064_7,
     author = {Hadwin, D. W.},
     title = {Dilations and {Hahn} {Decompositions} for {Linear} {Maps}},
     journal = {Canadian journal of mathematics},
     pages = {826--839},
     year = {1981},
     volume = {33},
     number = {4},
     doi = {10.4153/CJM-1981-064-7},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1981-064-7/}
}
TY  - JOUR
AU  - Hadwin, D. W.
TI  - Dilations and Hahn Decompositions for Linear Maps
JO  - Canadian journal of mathematics
PY  - 1981
SP  - 826
EP  - 839
VL  - 33
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1981-064-7/
DO  - 10.4153/CJM-1981-064-7
ID  - 10_4153_CJM_1981_064_7
ER  - 
%0 Journal Article
%A Hadwin, D. W.
%T Dilations and Hahn Decompositions for Linear Maps
%J Canadian journal of mathematics
%D 1981
%P 826-839
%V 33
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1981-064-7/
%R 10.4153/CJM-1981-064-7
%F 10_4153_CJM_1981_064_7

[1] 1. Arveson, W., Subalgebras of C*-algebras, Acta Math. 123 (1969), 141–224. Google Scholar

[2] 2. Barnes, B. A., The similarity problem for representations of a B*'-algebra, Michigan Math. J. 22 (1975), 25–32. Google Scholar

[3] 3. Berberian, S. K., Notes on spectral theory, Van Nostrand, Math. Studies No. 5 (Princeton, 1966). Google Scholar

[4] 4. Berberian, S. K., Approximate proper vectors, Proc. AMS 13 (1962), 111–114. Google Scholar

[5] 5. Bunce, J. W., Representations of strongly amenable C*-algebras, PAMS 32 (1972), 241–246. Google Scholar

[6] 6. Bunce, J. W., The similarity problem for representations of C*-algebras, preprint (1980). Google Scholar

[7] 7. Christensen, E., On non self-adjoint representations of C*-algebras, preprint (1980). Google Scholar

[8] 8. Dunford, N. and Schwartz, J. T., Linear operators I, II, III (Wiley, Interscience, New York). Google Scholar

[9] 9. Hadwin, D. W. and Nordgren, E. A., Subalgebras of reflexive algebras, preprint (1980). Google Scholar

[10] 10. Kadison, R. V., On the orthogonalization of operator representations, Amer. J. Math. 77 (1955), 600–620. Google Scholar

[11] 11. Loebl, R. I., A Hahn decomposition for linear maps, Pacific J. Math. 65 (1976), 119–133. Google Scholar

[12] 12. Loebl, R. I., Contractive linear maps on C*-algebras, Michigan Math. J. 22 (1975), 361–366. Google Scholar

[13] 13. Naimark, M. A., On a representation of operator set functions, Doklady Akad Nauk SSSR (N.S.) 41 (1943), 359–361. Google Scholar

[14] 14. Sakai, S., C*-algebras and W*-algebras, Ergebn. Math. Grenzgeb. 60 (Springer, Berlin, 1971). Google Scholar

[15] 15. Sarason, D. E., On spectral sets having connected complement, Acta. Sci. Math. (Szeged) 26 (1965), 289–299. Google Scholar

[16] 16. Stinespring, W. F., Positive functions on C*-algebras, PAMS 6 (1955), 211–216. Google Scholar

[17] 17. Tsui, S.-K., Decompositions of linear maps, TAMS 230 (1977), 87–112. Google Scholar

[18] 18. Wright, S., On orthogonalization of C*''-algebras, Indiana Univ. Math. J. 27 (1978), 383–399. Google Scholar

Cité par Sources :