The unique representation of a selfadjoint bounded linear functional

Suen, Ching Yun

doi:10.1090/S0002-9939-1985-0796446-4

The unique representation of a selfadjoint bounded linear functional
HTML articles powered by AMS MathViewer

by Ching Yun Suen PDF

Proc. Amer. Math. Soc. 95 (1985), 58-62 Request permission

Abstract:

It is well known that every selfadjoint bounded linear functional on a ${C^ * }$-algebra has a unique minimal decomposition [6, Theorem 3.2.5]. In this paper we prove that under some conditions a selfadjoint completely bounded linear map with a unique minimal decomposition is equivalent to the map with a unique commutant representation (up to unitary equivalence). Using the results, we generalize the Gel’fand-Naimark-Segal construction.

References

William B. Arveson, Subalgebras of $C^{\ast }$-algebras, Acta Math. 123 (1969), 141–224. MR 253059, DOI 10.1007/BF02392388
Man Duen Choi, Positive linear maps on $C^{\ast }$-algebras, Canadian J. Math. 24 (1972), 520–529. MR 315460, DOI 10.4153/CJM-1972-044-5
Man Duen Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl. 10 (1975), 285–290. MR 376726, DOI 10.1016/0024-3795(75)90075-0
Vern I. Paulsen, Every completely polynomially bounded operator is similar to a contraction, J. Funct. Anal. 55 (1984), no. 1, 1–17. MR 733029, DOI 10.1016/0022-1236(84)90014-4
Vern I. Paulsen and Ching Yun Suen, Commutant representations of completely bounded maps, J. Operator Theory 13 (1985), no. 1, 87–101. MR 768304
Gert K. Pedersen, $C^{\ast }$-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR 548006
Walter Rudin, Real and complex analysis, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210528
R. R. Smith, Completely bounded maps between $C^{\ast }$-algebras, J. London Math. Soc. (2) 27 (1983), no. 1, 157–166. MR 686514, DOI 10.1112/jlms/s2-27.1.157
W. Forrest Stinespring, Positive functions on $C^*$-algebras, Proc. Amer. Math. Soc. 6 (1955), 211–216. MR 69403, DOI 10.1090/S0002-9939-1955-0069403-4

The representation theory of completely bounded maps on

-algebras

Toshiyuki Takasaki and Jun Tomiyama, On the geometry of positive maps in matrix algebras, Math. Z. 184 (1983), no. 1, 101–108. MR 711731, DOI 10.1007/BF01162009
Jun Tomiyama, On the transpose map of matrix algebras, Proc. Amer. Math. Soc. 88 (1983), no. 4, 635–638. MR 702290, DOI 10.1090/S0002-9939-1983-0702290-4

Recent development of the theory of completely bounded maps between

-algebras