Positive operator bimeasures and a noncommutative generalization

Kari Ylinen

Studia Mathematica (1996)

  • Volume: 118, Issue: 2, page 157-168
  • ISSN: 0039-3223

Abstract

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For C*-algebras A and B and a Hilbert space H, a class of bilinear maps Φ: A× B → L(H), analogous to completely positive linear maps, is studied. A Stinespring type representation theorem is proved, and in case A and B are commutative, the class is shown to coincide with that of positive bilinear maps. As an application, the extendibility of a positive operator bimeasure to a positive operator measure is shown to be equivalent to various conditions involving positive scalar bimeasures, pairs of commuting projection-valued measures or pairs of commuting positive operator measures.

How to cite

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Ylinen, Kari. "Positive operator bimeasures and a noncommutative generalization." Studia Mathematica 118.2 (1996): 157-168. <http://eudml.org/doc/216270>.

@article{Ylinen1996,
abstract = {For C*-algebras A and B and a Hilbert space H, a class of bilinear maps Φ: A× B → L(H), analogous to completely positive linear maps, is studied. A Stinespring type representation theorem is proved, and in case A and B are commutative, the class is shown to coincide with that of positive bilinear maps. As an application, the extendibility of a positive operator bimeasure to a positive operator measure is shown to be equivalent to various conditions involving positive scalar bimeasures, pairs of commuting projection-valued measures or pairs of commuting positive operator measures.},
author = {Ylinen, Kari},
journal = {Studia Mathematica},
keywords = {-algebras; bilinear maps; completely positive linear maps; Stinespring type representation; positive operator bimeasure; positive operator measure; commuting projection-valued measures; pairs of commuting positive operator measures},
language = {eng},
number = {2},
pages = {157-168},
title = {Positive operator bimeasures and a noncommutative generalization},
url = {http://eudml.org/doc/216270},
volume = {118},
year = {1996},
}

TY - JOUR
AU - Ylinen, Kari
TI - Positive operator bimeasures and a noncommutative generalization
JO - Studia Mathematica
PY - 1996
VL - 118
IS - 2
SP - 157
EP - 168
AB - For C*-algebras A and B and a Hilbert space H, a class of bilinear maps Φ: A× B → L(H), analogous to completely positive linear maps, is studied. A Stinespring type representation theorem is proved, and in case A and B are commutative, the class is shown to coincide with that of positive bilinear maps. As an application, the extendibility of a positive operator bimeasure to a positive operator measure is shown to be equivalent to various conditions involving positive scalar bimeasures, pairs of commuting projection-valued measures or pairs of commuting positive operator measures.
LA - eng
KW - -algebras; bilinear maps; completely positive linear maps; Stinespring type representation; positive operator bimeasure; positive operator measure; commuting projection-valued measures; pairs of commuting positive operator measures
UR - http://eudml.org/doc/216270
ER -

References

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  1. [1] S. K. Berberian, Notes on Spectral Theory, Van Nostrand Math. Stud. 5, Van Nostrand, Princeton, N.J., 1966. Zbl0138.39104
  2. [2] C. Berg, J. P. R. Christensen and P. Ressel, Harmonic Analysis on Semigroups. Theory of Positive-Definite and Related Functions, Grad. Texts in Math. 100, Springer, New York, 1984. Zbl0619.43001
  3. [3] M. Sh. Birman, A. M. Vershik and M. Z. Solomyak, Product of commuting spectral measures need not be countably additive, Funktsional. Anal. i Prilozhen. 13 (1) (1978), 61-62; English transl.: Funct. Anal. Appl. 13 (1979), 48-49. Zbl0423.28008
  4. [4] P. D. Chen and J. F. Li, On the existence of product stochastic measures, Acta Math. Appl. Sinica 7 (1991), 120-134. Zbl0735.60001
  5. [5] E. Christensen and A. M. Sinclair, Representations of completely bounded multilinear operators, J. Funct. Anal. 72 (1987), 151-181. Zbl0622.46040
  6. [6] E. B. Davies, Quantum Theory of Open Systems, Academic Press, London, 1976. Zbl0388.46044
  7. [7] P. R. Halmos, Measure Theory, Van Nostrand, Toronto, 1950. 
  8. [8] A. S. Holevo, A noncommutative generalization of conditionally positive definite functions, in: Quantum Probability and Applications III (Proc. Conf. Oberwolfach, 1987), Lecture Notes in Math. 1303, Springer, Berlin, 1988, 128-148. 
  9. [9] S. Karni and E. Merzbach, On the extension of bimeasures, J. Anal. Math. 55 (1990), 1-16. Zbl0724.28003
  10. [10] C. Lance, On nuclear C*-algebras, J. Funct. Anal. 12 (1973), 157-176. 
  11. [11] M. Ozawa, Quantum measuring processes of continuous observables, J. Math. Phys. 25 (1984), 79-87. 
  12. [12] V. I. Paulsen, Completely Bounded Maps and Dilations, Pitman Res. Notes Math. 146, Longman, London, 1986. Zbl0614.47006
  13. [13] Z.-J. Ruan, The structure of pure completely bounded and completely positive multilinear operators, Pacific J. Math. 143 (1990), 155-173. Zbl0663.46050
  14. [14] W. F. Stinespring, Positive functions on C*-algebras, Proc. Amer. Math. Soc. 6 (1955), 211-216. Zbl0064.36703
  15. [15] M. Takesaki, Theory of Operator Algebras I, Springer, New York, 1979. 
  16. [16] K. Ylinen, On vector bimeasures, Ann. Mat. Pura Appl. 117 (1978), 115-138. Zbl0399.46032
  17. [17] K. Ylinen, Representations of bimeasures, Studia Math. 104 (1993), 269-278. Zbl0809.28001

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