Standard systems for semifinite O*-algebras

Inoue, Atsushi

doi:10.1090/S0002-9939-97-03962-2

Standard systems for semifinite O$^{*}$-algebras
HTML articles powered by AMS MathViewer

by Atsushi Inoue PDF

Proc. Amer. Math. Soc. 125 (1997), 3303-3312 Request permission

Abstract:

We shall continue the study of standard systems which make it possible to develop the Tomita-Takesaki theory in O$^*$-algebras. The main purpose of this paper is to give the necessary and sufficient conditions for which a standard system $(\mathcal {M}, \lambda , \lambda ’)$ of an O$^*$-algebra $\mathcal {M}$, a generalized vector $\lambda$ and the commutant $\lambda ’$ is unitarily equivalent to a standard system $\bigl ( \mathcal {N}, K’ \mu , (K’ \mu )’\bigr )$ constructed by a standard tracial generalized vector $\mu$ for an O$^*$-algebra $\mathcal {N}$ and a non-singular positive self-adjoint operator $K’$ affiliated with the commutant $\mathcal {N}’_{ \mathrm {w}}$ of $\mathcal {N}$.

References

J.-P. Antoine, H. Ogi, A. Inoue, and C. Trapani, Standard generalized vectors in the space of Hilbert-Schmidt operators, Ann. Inst. H. Poincaré Phys. Théor. 63 (1995), no. 2, 177–210 (English, with English and French summaries). MR 1357495
P. G. Dixon, Unbounded operator algebras, Proc. London Math. Soc. (3) 23 (1971), 53–69. MR 291821, DOI 10.1112/plms/s3-23.1.53
S. Gudder and W. Scruggs, Unbounded representations of $\ast$-algebras, Pacific J. Math. 70 (1977), no. 2, 369–382. MR 482269
Atsushi Inoue, On a class of unbounded operator algebras, Pacific J. Math. 65 (1976), no. 1, 77–95. MR 512382
Atsushi Inoue, An unbounded generalization of the Tomita-Takesaki theory, Publ. Res. Inst. Math. Sci. 22 (1986), no. 4, 725–765. MR 871264, DOI 10.2977/prims/1195177629
Atsushi Inoue, Standard generalized vectors for algebras of unbounded operators, J. Math. Soc. Japan 47 (1995), no. 2, 329–347. MR 1317285, DOI 10.2969/jmsj/04720329
Atsushi Inoue and Witold Karwowski, Cyclic generalized vectors for algebras of unbounded operators, Publ. Res. Inst. Math. Sci. 30 (1994), no. 4, 577–601. MR 1308958, DOI 10.2977/prims/1195165790
G. Lassner, Topological algebras of operators, Rep. Mathematical Phys. 3 (1972), no. 4, 279–293. MR 322527, DOI 10.1016/0034-4877(72)90012-2
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
Robert T. Powers, Self-adjoint algebras of unbounded operators, Comm. Math. Phys. 21 (1971), 85–124. MR 283580
Robert T. Powers, Algebras of unbounded operators, Operator algebras and applications, Part 2 (Kingston, Ont., 1980) Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 389–406. MR 679528
Marc A. Rieffel and Alfons van Daele, A bounded operator approach to Tomita-Takesaki theory, Pacific J. Math. 69 (1977), no. 1, 187–221. MR 438147
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Konrad Schmüdgen, Unbounded operator algebras and representation theory, Operator Theory: Advances and Applications, vol. 37, Birkhäuser Verlag, Basel, 1990. MR 1056697, DOI 10.1007/978-3-0348-7469-4
M. Takesaki, Tomita’s theory of modular Hilbert algebras and its applications, Lecture Notes in Mathematics, Vol. 128, Springer-Verlag, Berlin-New York, 1970. MR 0270168
Alfons van Daele, A new approach to the Tomita-Takesaki theory of generalized Hilbert algebras, J. Functional Analysis 15 (1974), 378–393. MR 0346539, DOI 10.1016/0022-1236(74)90029-9