Regular weights on algebras of unbounded operators

Atsushi INOUE; Hidekazu OGI

doi:10.2969/jmsj/05010227

Atsushi INOUE, Hidekazu OGI

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1998 年 50 巻 1 号 p. 227-252

DOI https://doi.org/10.2969/jmsj/05010227

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Published: 1998 Received: 1995/09/08 Available on J-STAGE: 2006/10/20 Accepted: - Advance online publication: - Revised: -
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訂正日: 2006/10/20 訂正理由: - 訂正箇所: 論文受付日訂正内容: 訂正前 : 19950905
訂正後 : 19950908

訂正日: 2006/10/20 訂正理由: - 訂正箇所: 引用文献情報訂正内容: Wrong : 1) J. P. Antoine and A. lnoue, Representability of invariant positive sesquilinear forms on partial *-algebras, Math. Proc. Camb. Phil. Soc., 108 (1990), 337-353.
2) J. P. Antoine and A. Inoue, Normal forms on partial O^*-algebras, J. Math. Phys., 32 (1991), 2074-2081.
3) J. P. Antoine, Y. Soulet and C. Trapani, Weights on partial *-algebras, J. Math. Anal. Appl., (to appear).
4) J. P. Antoine, A. Inoue, H. Ogi and C. Trapani, Standard generalized vectors in the space of Hilbert-Schmidt operators, Ann. Inst. H. Poincaré (to appear).
5) H. J. Borchers, Algebraic aspects of Wightman field theory, In Statistical Mechanics and Field Theory, Lecture, 1971; Haifa Summer School, New York-Jerusalem-London (1972), 31-79.
6) O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics I, II, Springer Verlag, New York, Heidelberg, Berlin, 1979, 1981.
7) S. P. Gudder and W. Scruggs, Unbounded representations of *-algebras, Pacific J. Math., 70 (1977), 369-382.
8) S. P. Gudder and R. L. Hudson, A noncommutative probability theory, Trans. Amer. Math. Soc., 245 (1978), 1-41.
9) U. Haagerup, Normal weights on W^*-algebras, J. Functional Anal. 19 (1975), 305-317.
10) A. Inoue, Regularity of positive linear functionals, Japanese J. Math., 9 (1983), 247-275.
11) A. Inoue, Modular structure of algebras of unbounded operators, Math. Proc. Camb. Phil. Soc., 111 (1992), 369-386.
12) A. Inoue, Modular systems induced by trace functionals on algebras of unbounded operators, J. Math. Phys., 35 (1994), 435-442.
13) A. Inoue, O^*-algebras in standard system, Math. Nachr., 172 (1995), 171-190.
14) A. Inoue, Standard generalized vectors for algebras of unbounded operators, J. Math. Soc. Japan., 47 (1995), 329-347.
15) A. Inoue and W. Karwowski, Cyclic generalized vectors for algebras of unbounded operators, Publ. RIMS, Kyoto Univ., 30 (1994), 577-601.
16) P. E. T. Jorgensen, Operators and Representation Theory, North-Holland, Amsterdam, 1988.
17) K. Katavolos and I. Koch, Extension of Tomita-Takesaki theory to the unbounded algebra of the canonical commutation relation, Rep. Math. Phys., 16 (1979), 335-352.
18) G. Lassner, Topological algebras of operators, Rep. Math. Phys., 3 (1972), 279-293.
19) G. Lassner, Algebras of unbounded operators and quantum dynamics, Physica, 124 A (1984), 471-480.
20) G. Lassner, Topological algebras and their applications in quantum statistics, Wiss. Z. Karl-Marx-Univ. Leipzig, Math. Naturwiss. R., 30 (1981), 572-595.
21) G. Lassner and W. Timmermann, Normal states on algebras of unbounded operators, Rep. Math. Phys., 3 (1972), 295-305.
22) R. T. Powers, Self-adjoint algebras of unbounded operators, Commun. Math. Phys., 21, (1971), 85-124.
23) R. T. Powers, Algebras of unbounded operators, Proc. Sym. Pure Math., 38 (1982), 389-406.
24) K. Schmüdgen, The order structure of topological *-algebras of unbounded operators, Rep. Math. Phys., 7 (1975), 215-227.
25) K. Schmüdgen, An example of a positive polynomial which is not a sum of squares of polynomials. A positive, but not strongly positive functional, Math. Nachr., 88 (1979), 385-390.
26) K. Schmüdgen, On trace representation of linear functionals on unbounded operator algebras, Commun. Math. Phys., 63 (1978), 113-130.
27) K. Schmüdgen, A proof of a theorem on trace representation of strongly positive linear functionals on O^*_p-algebras, J. Operator Theory, 2 (1979), 39-47.
28) K. Schmüdgen, Unbounded Operator Algebras and Representation Theory, Akademie-Verlag Berlin,1990.
29) T. Scherman, Positive linear functionals on *-algebras of unbounded operators, J. Math. Anal. Appl., 22 (1968), 285-318.
30) S. Stratila and L. Zsido, Lectures on von Neumann algebras, Abacus Press, Tunbridge Wells, 1979.
31) S. Stratila, Modular Theory in Operator Algebras, Abacus Press, Tunbridge Wells, 1981.
32) W. Thirring and A. Wehrl, On the mathematical structure of the B.S.C.-model, Commun. Math. Phys., 4 (1967), 303-314.
33) W. Thirring, On the mathematical stucture of the B.S.C.-model. II, Commun. Math. Phys., 7 (1968), 181-189.
34) S. L. Woronowicz, The quantum problem of moments I, Rep. Math. Phys., I (1970), 135-145; II, Rep. Math. Phys., I (1971), 175-183.

Right : [1] J. P. Antoine and A. Inoue, Representability of invariant positive sesquilinear forms on partial *-algebras, Math. Proc. Camb. Phil. Soc., 108 (1990), 337-353.
[2] J. P. Antoine and A. Inoue, Normal forms on partial O^*-algebras, J. Math. Phys., 32 (1991), 2074-2081.
[3] J. P. Antoine, Y. Soulet and C. Trapani, Weights on partial *-algebras, J. Math. Anal. Appl., (to appear).
[4] J. P. Antoine, A. Inoue, H. Ogi and C. Trapani, Standard generalized vectors in the space of Hilbert-Schmidt operators, Ann. Inst. H. Poincaré (to appear).
[5] H. J. Borchers, Algebraic aspects of Wightman field theory, In Statistical Mechanics and Field Theory, Lecture, 1971; Haifa Summer School, New York-Jerusalem-London (1972), 31-79.
[6] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics I, II, Springer Verlag, New York, Heidelberg, Berlin, 1979, 1981.
[7] S. P. Gudder and W. Scruggs, Unbounded representations of *-algebras, Pacific J. Math., 70 (1977), 369-382.
[8] S. P. Gudder and R. L. Hudson, A noncommutative probability theory, Trans. Amer. Math. Soc., 245 (1978), 1-41.
[9] U. Haagerup, Normal weights on W^*-algebras, J. Functional Anal. 19 (1975), 305-317.
[10] A. Inoue, Regularity of positive linear functionals, Japanese J. Math., 9 (1983), 247-275.
[11] A. Inoue, Modular structure of algebras of unbounded operators, Math. Proc. Camb. Phil. Soc., 111 (1992), 369-386.
[12] A. Inoue, Modular systems induced by trace functionals on algebras of unbounded operators, J. Math. Phys., 35 (1994), 435-442.
[13] A. Inoue, O^*-algebras in standard system, Math. Nachr., 172 (1995), 171-190.
[14] A. Inoue, Standard generalized vectors for algebras of unbounded operators, J. Math. Soc. Japan., 47 (1995), 329-347.
[15] A. Inoue and W. Karwowski, Cyclic generalized vectors for algebras of unbounded operators, Publ. RIMS, Kyoto Univ., 30 (1994), 577-601.
[16] P. E. T. Jorgensen, Operators and Representation Theory, North-Holland, Amsterdam, 1988.
[17] K. Katavolos and I. Koch, Extension of Tomita-Takesaki theory to the unbounded algebra of the canonical commutation relation, Rep. Math. Phys., 16 (1979), 335-352.
[18] G. Lassner, Topological algebras of operators, Rep. Math. Phys., 3 (1972), 279-293.
[19] G. Lassner, Algebras of unbounded operators and quantum dynamics, Physica, 124 A (1984), 471-480.
[20] G. Lassner, Topological algebras and their applications in quantum statistics, Wiss. Z. Karl-Marx-Univ. Leipzig, Math. Naturwiss. R., 30 (1981), 572-595.
[21] G. Lassner and W. Timmermann, Normal states on algebras of unbounded operators, Rep. Math. Phys., 3 (1972), 295-305.
[22] R. T. Powers, Self-adjoint algebras of unbounded operators, Commun. Math. Phys., 21, (1971), 85-124.
[23] R. T. Powers, Algebras of unbounded operators, Proc. Sym. Pure Math., 38 (1982), 389-406.
[24] K. Schmüdgen, The order structure of topological *-algebras of unbounded operators, Rep. Math. Phys., 7 (1975), 215-227.
[25] K. Schmüdgen, An example of a positive polynomial which is not a sum of squares of polynomials. A positive, but not strongly positive functional, Math. Nachr., 88 (1979), 385-390.
[26] K. Schmüdgen, On trace representation of linear functionals on unbounded operator algebras, Commun. Math. Phys., 63 (1978), 113-130.
[27] K. Schmüdgen, A proof of a theorem on trace representation of strongly positive linear functionals on O^*_p-algebras, J. Operator Theory, 2 (1979), 39-47.
[28] K. Schmüdgen, Unbounded Operator Algebras and Representation Theory, Akademie-Verlag Berlin,1990.
[29] T. Scherman, Positive linear functionals on *-algebras of unbounded operators, J. Math. Anal. Appl., 22 (1968), 285-318.
[30] S. Stratila and L. Zsido, Lectures on von Neumann algebras, Abacus Press, Tunbridge Wells, 1979.
[31] S. Stratila, Modular Theory in Operator Algebras, Abacus Press, Tunbridge Wells, 1981.
[32] W. Thirring and A. Wehrl, On the mathematical structure of the B. S. C. -model, Commun. Math. Phys., 4 (1967), 303-314.
[33] W. Thirring, On the mathematical stucture of the B. S. C. -model. II, Commun. Math. Phys., 7 (1968), 181-189.
[34] S. L. Woronowicz, The quantum problem of moments I, Rep. Math. Phys., I (1970), 135-145; II, Rep. Math. Phys., I (1971), 175-183.

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