Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 9 (2014), 149 -- 166

A COVARIANT STINESPRING TYPE THEOREM FOR τ-MAPS

Harsh Trivedi

Abstract. Let τ be a linear map from a unital C*-algebra A to a von Neumann algebra B and let C be a unital C*-algebra. A map T from a Hilbert A-module E to a von Neumann C-B module F is called a τ-map if

=τ( ) for all x,y∈ E.
A Stinespring type theorem for τ-maps and its covariant version are obtained when τ is completely positive. We show that there is a bijective correspondence between the set of all τ-maps from E to F which are (u',u)-covariant with respect to the dynamical system (G,η,E) and the set of all (u' ,u)-covariant τ~-maps from the crossed product E×η G to F, where τ and τ~ are completely positive.

2010 Mathematics Subject Classification: Primary: 46L08, 46L55; Secondary: 46L07, 46L53.
Keywords: Stinespring representation; Completely positive maps; Von Neumann modules; Dynamical systems.

Full text

References

  1. Panchugopal Bikram, Kunal Mukherjee, R. Srinivasan, and V. S. Sunder, Hilbert von Neumann modules, Commun. Stoch. Anal. 6 (2012), no. 1, 49--64. MR2890849

  2. B. V. Rajarama Bhat, G. Ramesh, and K. Sumesh, Stinespring's theorem for maps on Hilbert C*-modules, J. Operator Theory 68 (2012), no. 1, 173--178. MR 2966040. Zbl 1265.46091.

  3. B. V. Rajarama Bhat and K. Sumesh, Bures distance for completely positive maps, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16 (2013), no. 4, 1350031, 22. MR 3192708. Zbl 1295.46043.

  4. Man Duen Choi and Edward G. Effros, The completely positive lifting problem for C*-algebras, Ann. of Math. (2) 104 (1976), no. 3, 585--609. MR 0417795 (54 #5843). Zbl 0361.46067.

  5. Santanu Dey and Harsh Trivedi, \mathfrakK-families and CPD-H-extendable families, arXiv:1409.3655v1 (2014). arXiv:1409.3655v1.

  6. Siegfried Echterhoff, S. Kaliszewski, John Quigg, and Iain Raeburn, Naturality and induced representations, Bull. Austral. Math. Soc. 61 (2000), no. 3, 415--438. MR 1762638 (2001j:46101). Zbl 0982.46051.

  7. Heo Jaeseong, Completely multi-positive linear maps and representations on Hilbert C*-modules, J. Operator Theory 41 (1999), no. 1, 3--22. MR 1675235. Zbl 0994.46019.

  8. Heo Jaeseong and Ji Un Cig, Quantum stochastic processes for maps on Hilbert C*-modules, J. Math. Phys. 52 (2011), no. 5, 053501, 16pp. MR 2839082.

  9. Maria Joiţa, Covariant version of the Stinespring type theorem for Hilbert C*-modules, Cent. Eur. J. Math. 9 (2011), no. 4, 803--813. MR 2805314 (2012f:46110). Zbl 1243.46049.

  10. Maria Joiţa, Covariant representations of Hilbert C*-modules, Expo. Math. 30 (2012), no. 2, 209--220. MR 2928201. Zbl 1252.46055.

  11. Alexander Kaplan, Covariant completely positive maps and liftings, Rocky Mountain J. Math. 23 (1993), no. 3, 939--946. MR 1245456 (94k:46139). Zbl 0796.46042.

  12. E. C. Lance, Hilbert C*-modules, London Mathematical Society Lecture Note Series, vol. 210, Cambridge University Press, Cambridge, 1995, A toolkit for operator algebraists. MR 1325694 (96k:46100). Zbl 0822.46080.

  13. William L. Paschke, Inner product modules over B* -algebras, Trans. Amer. Math. Soc. 182 (1973), 443--468. MR 0355613 (50 #8087). Zbl 0239.46062.

  14. Vern Paulsen, A covariant version of Ext, Michigan Math. J. 29 (1982), no. 2, 131--142. MR 654474 (83f:46076). Zbl 0507.46060.

  15. Marc A. Rieffel, Induced representations of C* -algebras, Advances in Math. 13 (1974), 176--257. MR 0353003 (50 #5489). Zbl 0284.46040.

  16. Iain Raeburn and Shaun J. Thompson, Countably generated Hilbert modules, the Kasparov stabilisation theorem, and frames with Hilbert modules, Proc. Amer. Math. Soc. 131 (2003), no. 5, 1557--1564 (electronic). MR 1949886 (2003j:46089). Zbl 1015.46034.

  17. Iain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace C*-algebras, Mathematical Surveys and Monographs, vol. 60, American Mathematical Society, Providence, RI, 1998. MR 1634408 (2000c:46108) Zbl 0922.46050.

  18. Michael Skeide, Generalised matrix C*-algebras and representations of Hilbert modules, Math. Proc. R. Ir. Acad. 100A (2000), no. 1, 11--38. MR 1882195 (2002k:46155). Zbl 0972.46038.

  19. Michael Skeide, Hilbert modules and applications in quantum probability, Habilitationsschrift (2001). skeide.html">Habilitationsschrift.

  20. Michael Skeide, Hilbert von Neumann modules versus concrete von Neumann modules, arXiv:1205.6413v1 (2012). arXiv:1205.6413v1.

  21. Skeide Michael and Sumesh K., CP-H-extendable maps between Hilbert modules and CPH-semigroups, J. Math. Anal. Appl. 414 (2014), no. 2, 886--913. 3168002.

  22. W. Forrest Stinespring, Positive functions on C*-algebras, Proc. Amer. Math. Soc. 6 (1955), 211--216. 0069403 (16,1033b). Zbl 0064.36703.

  23. Dana P. Williams, Crossed products of C*-algebras, Mathematical Surveys and Monographs, vol. 134, American Mathematical Society, Providence, RI, 2007. 2288954 (2007m:46003). Zbl 1119.46002.



Harsh Trivedi
Department of Mathematics, Indian Institute of Technology Bombay,
Powai, Mumbai-400076,
India.
E-mail: harsh@math.iitb.ac.in

http://www.utgjiu.ro/math/sma