Pure states on Jordan algebras

Jan Hamhalter

Mathematica Bohemica (2001)

  • Volume: 126, Issue: 1, page 81-91
  • ISSN: 0862-7959

Abstract

top
We prove that a pure state on a C * -algebras or a JB algebra is a unique extension of some pure state on a singly generated subalgebra if and only if its left kernel has a countable approximative unit. In particular, any pure state on a separable JB algebra is uniquely determined by some singly generated subalgebra. By contrast, only normal pure states on JBW algebras are determined by singly generated subalgebras, which provides a new characterization of normal pure states. As an application we contribute to the extension problem and strengthen the hitherto known results on independence of operator algebras arising in the quantum field theory.

How to cite

top

Hamhalter, Jan. "Pure states on Jordan algebras." Mathematica Bohemica 126.1 (2001): 81-91. <http://eudml.org/doc/248840>.

@article{Hamhalter2001,
abstract = {We prove that a pure state on a $C^\{\ast \}$-algebras or a JB algebra is a unique extension of some pure state on a singly generated subalgebra if and only if its left kernel has a countable approximative unit. In particular, any pure state on a separable JB algebra is uniquely determined by some singly generated subalgebra. By contrast, only normal pure states on JBW algebras are determined by singly generated subalgebras, which provides a new characterization of normal pure states. As an application we contribute to the extension problem and strengthen the hitherto known results on independence of operator algebras arising in the quantum field theory.},
author = {Hamhalter, Jan},
journal = {Mathematica Bohemica},
keywords = {JB algebras; $C^\{\ast \}$-algebras; pure states; state space independence of Jordan algebras; normal pure states on JBW algebras; JB algebras; -algebras; pure states; state space independence of Jordan algebras; normal pure states on JBW algebras},
language = {eng},
number = {1},
pages = {81-91},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Pure states on Jordan algebras},
url = {http://eudml.org/doc/248840},
volume = {126},
year = {2001},
}

TY - JOUR
AU - Hamhalter, Jan
TI - Pure states on Jordan algebras
JO - Mathematica Bohemica
PY - 2001
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 126
IS - 1
SP - 81
EP - 91
AB - We prove that a pure state on a $C^{\ast }$-algebras or a JB algebra is a unique extension of some pure state on a singly generated subalgebra if and only if its left kernel has a countable approximative unit. In particular, any pure state on a separable JB algebra is uniquely determined by some singly generated subalgebra. By contrast, only normal pure states on JBW algebras are determined by singly generated subalgebras, which provides a new characterization of normal pure states. As an application we contribute to the extension problem and strengthen the hitherto known results on independence of operator algebras arising in the quantum field theory.
LA - eng
KW - JB algebras; $C^{\ast }$-algebras; pure states; state space independence of Jordan algebras; normal pure states on JBW algebras; JB algebras; -algebras; pure states; state space independence of Jordan algebras; normal pure states on JBW algebras
UR - http://eudml.org/doc/248840
ER -

References

top
  1. 10.1090/S0002-9939-1969-0240633-1, Proc. Amer. Math. Soc. 21 (1969), 749–752. (1969) MR0240633DOI10.1090/S0002-9939-1969-0240633-1
  2. Approaching infinity in C * -algebras, J. Operator Theory 21 (1989), 255–271. (1989) MR1023315
  3. 10.2140/pjm.1970.33.543, Pacific J. Math., 33 (1970), 543–550. (1970) Zbl0184.16903MR0264406DOI10.2140/pjm.1970.33.543
  4. 10.1215/S0012-7094-68-03553-9, Duke Math. J. 35 (1968), 525–533. (1968) Zbl0172.41201MR0229048DOI10.1215/S0012-7094-68-03553-9
  5. Extensions, restrictions, and representations of states on C * -algebras, Trans. Amer. Math. Soc. 249 (1979), 303–323. (1979) Zbl0408.46049MR0525675
  6. 10.1016/0022-1236(79)90061-2, J. Func. Anal. 31 (1979), 195–217. (1979) MR0525951DOI10.1016/0022-1236(79)90061-2
  7. A maximal abelian subalgebra of the Calcin algebra with the extension property, Math. Scand. (1978), 101–110. (1978) MR0500149
  8. A conjecture concerning the pure states of B ( H ) and related theorem, In Topics in modern operator theory (Timisoara/Herculane, 1980), Birkhäuser, Basel-Boston, Mass. (1981), 27–43. (1981) MR0672813
  9. 10.1090/S0002-9939-1972-0295089-X, Proc. Amer. Math. Soc. 33 (1972), 491–494. (1972) Zbl0243.46060MR0295089DOI10.1090/S0002-9939-1972-0295089-X
  10. Characters on singly generated C * -algebras, Proc. Amer. Math. Soc. 25 (1970), 297–303. (1970) Zbl0195.42006MR0259622
  11. On the statistical independence of algebras of observables, J. Math. Phys. 3 (1997), 1318–1328. (1997) MR1435671
  12. Measures on the closed subspaces of a Hilbert space, J. Math. Mech. 6 (1957), 885–893. (1957) Zbl0078.28803MR0096113
  13. 10.2140/pjm.1960.10.547, Pacific J. Math. 10 (1960), 547–556. (1960) MR0115104DOI10.2140/pjm.1960.10.547
  14. Statistical independence of operator algebras, Ann. Inst. Henri Poincaré, 67 (1997), 447–462. (1997) Zbl0893.46048MR1632248
  15. 10.1090/S0002-9939-99-04919-9, Proc. Amer. Math. Soc. 127 (1999), 131–137. (1999) Zbl0907.46052MR1610905DOI10.1090/S0002-9939-99-04919-9
  16. Jordan Operator Algebras, Pitman Publishing, Boston, London, Melbourne, 1984. (1984) MR0755003
  17. Foundations of Quantum Mechanics, Addison Wesley, 1968. (1968) Zbl0166.23301MR0218062
  18. 10.1073/pnas.43.3.273, Proceeding of the National Academy of Sciences (U.S.A) 43 (1957), 273–276. (1957) Zbl0078.11502MR0085484DOI10.1073/pnas.43.3.273
  19. 10.2307/2372748, American J. Math. 81 (1959), 383–400. (1959) MR0123922DOI10.2307/2372748
  20. Mathematical Foundations of Quantum Mechanics, Benjamin, New York, 1963. (1963) Zbl0114.44002
  21. States and composite systems in W * -algebraic quantum mechanics, Diss. ETH, No. 6824, Zurich, 1981. (1981) 
  22. Independence of local algebras in quantum field theory, Commun. Math. Phys. 13 (1969), 216–225. (1969) MR0266539
  23. 10.1090/S0002-9947-1968-0217611-5, Trans. Amer. Math. Soc. 130 (1968), 153–166. (1968) MR0217611DOI10.1090/S0002-9947-1968-0217611-5
  24. A characterization of pure states of C * -algebras, Proc. Amer. Math. Soc. 19 (1968), 1100–1102. (1968) MR0232222
  25. On the independence of local algebras in quantum field theory, Reviews in Mathematical Physics 2 (1990), 201–247. (1990) Zbl0743.46079MR1090281

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.