| Title:
|
Pure states on Jordan algebras (English) |
| Author:
|
Hamhalter, Jan |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
126 |
| Issue:
|
1 |
| Year:
|
2001 |
| Pages:
|
81-91 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
JB algebras |
| Keyword:
|
$C^{\ast}$-algebras |
| Keyword:
|
pure states |
| Keyword:
|
state space independence of Jordan algebras |
| Keyword:
|
normal pure states on JBW algebras |
| MSC:
|
17C65 |
| MSC:
|
46H70 |
| MSC:
|
46L30 |
| MSC:
|
46L70 |
| MSC:
|
81P10 |
| idZBL:
|
Zbl 0983.46046 |
| idMR:
|
MR1826473 |
| DOI:
|
10.21136/MB.2001.133911 |
| . |
| Date available:
|
2009-09-24T21:47:38Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/133911 |
| . |
| Reference:
|
[1] J. F. Aarnes, R. V. Kadison: Pure states and approximate identities.Proc. Amer. Math. Soc. 21 (1969), 749–752. MR 0240633, 10.1090/S0002-9939-1969-0240633-1 |
| Reference:
|
[2] C. A. Akemann, J. Anderson, G. K. Pedersen: Approaching infinity in $C^\ast $-algebras.J. Operator Theory 21 (1989), 255–271. MR 1023315 |
| Reference:
|
[3] C. A. Akemann: Approximate units and maximal abelian $C^\ast $-subalgebras.Pacific J. Math., 33 (1970), 543–550. Zbl 0184.16903, MR 0264406, 10.2140/pjm.1970.33.543 |
| Reference:
|
[4] C. A. Akemann: Interpolation in $W^\ast $-algebras.Duke Math. J. 35 (1968), 525–533. Zbl 0172.41201, MR 0229048, 10.1215/S0012-7094-68-03553-9 |
| Reference:
|
[5] J. Anderson: Extensions, restrictions, and representations of states on $C^\ast $-algebras.Trans. Amer. Math. Soc. 249 (1979), 303–323. Zbl 0408.46049, MR 0525675 |
| Reference:
|
[6] J. Anderson: Extreme points in sets of positive linear maps on $B(H)$.J. Func. Anal. 31 (1979), 195–217. MR 0525951, 10.1016/0022-1236(79)90061-2 |
| Reference:
|
[7] J. Anderson: A maximal abelian subalgebra of the Calcin algebra with the extension property.Math. Scand. (1978), 101–110. MR 0500149 |
| Reference:
|
[8] J. Anderson: 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. MR 0672813 |
| Reference:
|
[9] B. A. Barnes: Pure states with the restriction property.Proc. Amer. Math. Soc. 33 (1972), 491–494. Zbl 0243.46060, MR 0295089, 10.1090/S0002-9939-1972-0295089-X |
| Reference:
|
[10] J. Bunce: Characters on singly generated $C^\ast $-algebras.Proc. Amer. Math. Soc. 25 (1970), 297–303. Zbl 0195.42006, MR 0259622 |
| Reference:
|
[11] M. Floring, S. J. Summers: On the statistical independence of algebras of observables.J. Math. Phys. 3 (1997), 1318–1328. MR 1435671 |
| Reference:
|
[12] A. M. Gleason: Measures on the closed subspaces of a Hilbert space.J. Math. Mech. 6 (1957), 885–893. Zbl 0078.28803, MR 0096113 |
| Reference:
|
[13] J. G. Glimm, R. V. Kadison: Unitary operators in $C^\ast $-algebras.Pacific J. Math. 10 (1960), 547–556. MR 0115104, 10.2140/pjm.1960.10.547 |
| Reference:
|
[14] J. Hamhalter: Statistical independence of operator algebras.Ann. Inst. Henri Poincaré, 67 (1997), 447–462. Zbl 0893.46048, MR 1632248 |
| Reference:
|
[15] J. Hamhalter: Universal state space embeddability of Jordan-Banach algebras.Proc. Amer. Math. Soc. 127 (1999), 131–137. Zbl 0907.46052, MR 1610905, 10.1090/S0002-9939-99-04919-9 |
| Reference:
|
[16] H.-Olsen. Hanche, E. Stormer: Jordan Operator Algebras.Pitman Publishing, Boston, London, Melbourne, 1984. MR 0755003 |
| Reference:
|
[17] J. M. Jauch: Foundations of Quantum Mechanics.Addison Wesley, 1968. Zbl 0166.23301, MR 0218062 |
| Reference:
|
[18] R. V. Kadison: Irreducible operator algebras.Proceeding of the National Academy of Sciences (U.S.A) 43 (1957), 273–276. Zbl 0078.11502, MR 0085484, 10.1073/pnas.43.3.273 |
| Reference:
|
[19] R. V. Kadison, I. M Singer: Extensions of pure states.American J. Math. 81 (1959), 383–400. MR 0123922, 10.2307/2372748 |
| Reference:
|
[20] G. W. Mackey: Mathematical Foundations of Quantum Mechanics.Benjamin, New York, 1963. Zbl 0114.44002 |
| Reference:
|
[21] G. A. Raggio: States and composite systems in $W^\ast $-algebraic quantum mechanics.Diss. ETH, No. 6824, Zurich, 1981. |
| Reference:
|
[22] H. Roos: Independence of local algebras in quantum field theory.Commun. Math. Phys. 13 (1969), 216–225. MR 0266539 |
| Reference:
|
[23] E. Stormer: Irreducible Jordan algebras of self-adjoint operators.Trans. Amer. Math. Soc. 130 (1968), 153–166. MR 0217611, 10.1090/S0002-9947-1968-0217611-5 |
| Reference:
|
[24] E. Stormer: A characterization of pure states of $C^\ast $-algebras.Proc. Amer. Math. Soc. 19 (1968), 1100–1102. MR 0232222 |
| Reference:
|
[25] S. J. Summers: On the independence of local algebras in quantum field theory.Reviews in Mathematical Physics 2 (1990), 201–247. Zbl 0743.46079, MR 1090281 |
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