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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  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
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  9. [9] S. Karni and E. Merzbach, On the extension of bimeasures, J. Anal. Math. 55 (1990), 1-16. Zbl0724.28003
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  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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