Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Tracial positive linear maps of $ C\sp{\ast} $-algebras


Authors: Man Duen Choi and Sze Kai Tsui
Journal: Proc. Amer. Math. Soc. 87 (1983), 57-61
MSC: Primary 46L05
DOI: https://doi.org/10.1090/S0002-9939-1983-0677231-9
MathSciNet review: 677231
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A positive linear map $ \Phi :\mathfrak{A} \to \mathfrak{B}$ between two $ {C^ * }$-algebras is said to be tracial if $ \Phi ({A_1}{A_2}) = \Phi ({A_2}{A_1})$ for all $ {A_i} \in \mathfrak{A}$. A tracial positive linear map $ \mathfrak{A} \to \mathcal{B}\left(\mathcal{H} \right)$ is analyzed as the composition of a tracial positive linear map $ \mathfrak{A} \to C(X)$ followed by a positive linear map $ C(X) \to \mathcal{B}\left( \mathcal{H} \right)$.


References [Enhancements On Off] (What's this?)

  • [1] W. B. Arveson, Notes on extensions of $ {C^ * }$-algebras, Duke Math. J. 44 (1977), 329-355. MR 0438137 (55:11056)
  • [2] M. D. Choi, A Schwarz inequality for positive linear maps on $ {C^ * }$-algebras, Illinois J. Math. 18 (1974), 565-574. MR 0355615 (50:8089)
  • [3] -, Some assorted inequalities for positive linear maps on $ {C^ * }$-algebras, J. Operator Theory 4 (1980), 271-285. MR 595415 (82c:46073)
  • [4] -, Positive linear maps, Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R. I. (to appear).
  • [5] M. D. Choi and E. G. Effros, The completely positive lifting problem for $ {C^ * }$-algebras, Ann. of Math. (2) 104 (1976), 585-609. MR 0417795 (54:5843)
  • [6] J. Dixmier, Les algèbres d'opérateurs dans l'espace hilbertien, Gauthier-Villars, Paris, 1957. MR 0094722 (20:1234)
  • [7] -, $ {C^ * }$-algebras, North-Holland, Amsterdam, 1977.
  • [8] R. G. Douglas, Banach algebra techniques in operator theory, Academic Press, New York, 1972. MR 0361893 (50:14335)
  • [9] L. T. Gardner, Linear maps of $ {C^ * }$-algebras preserving the absolute value, Proc. Amer. Math. Soc. 76 (1979), 271-278. MR 537087 (80h:46090)
  • [10] E. Størmer, Positive linear maps of operator algebras, Acta Math. 110 (1963), 233-278. MR 0156216 (27:6145)
  • [11] -, Positive linear maps of $ {C^ * }$-algebras, Lecture Notes in Physics, Vol. 29, Springer-Verlag, Berlin and New York, 1974, pp. 85-106.
  • [12] J. Vesterstrøm, Positive linear extension operators for space of affine functions, Israel J. Math. 16 (1973), 203-211. MR 0343005 (49:7749)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46L05

Retrieve articles in all journals with MSC: 46L05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0677231-9
Keywords: $ {C^ * }$-algebras, finite von Neumann algebras, positive linear maps, traces, Schwarz inequality, Toeplitz operators
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society