Arens spaces associated with von Neumann algebras and normal states. (English) Zbl 1200.46054
Summary: We introduce the notion of Arens space associated with a finite von Neumann algebra and a faithful normal finite state. Relations between these spaces and Arens algebras with respect to traces are investigated.
MSC:
| 46L51 | Noncommutative measure and integration |
| 46L52 | Noncommutative function spaces |
| 46L07 | Operator spaces and completely bounded maps |
References:
| [1] | Abdullaev R.Z.: Isomorphism of Arens algebras. Siberian J. Ind. Math. 1(2), 3–13 (1998) · Zbl 0959.46046 |
| [2] | Albeverio S., Ayupov Sh.A., Kudaybergenov K.K.: Non commutative Arens algebras and their derivations. J. Funct. Anal. 253(1), 287–302 (2007) · Zbl 1138.46040 · doi:10.1016/j.jfa.2007.04.010 |
| [3] | Arens R.: The space L {\(\omega\)} (0; 1) and convex topological rings. Bull. Am. Math. Soc. 52, 931–935 (1946) · Zbl 0060.26901 · doi:10.1090/S0002-9904-1946-08681-4 |
| [4] | Dixmier J.: Formes linéaires sur un anneau d’opérateurs. Bull. Soc. Math. France 81(1), 9–39 (1953) · Zbl 0050.11501 |
| [5] | Inoue A.: On a class of unbounded operators II. Pacific J. Math. 66, 411–431 (1976) · Zbl 0355.46034 |
| [6] | Pedersen G.K., Takesaki M.: The Radon–Nikodym theorem for von Neumann algebras. Acta. Math. 130(1–2), 53–87 (1973) · Zbl 0262.46063 · doi:10.1007/BF02392262 |
| [7] | Segal I.: A non-commutative extension of abstract integration. Ann. Math. 57, 401–457 (1953) · Zbl 0051.34201 · doi:10.2307/1969729 |
| [8] | Takesaki, M.: Theory of Operator Algebras. I. Springer, New York, p. XII+415 (1979) · Zbl 0436.46043 |
| [9] | Trunov N.V.: On a noncommutative analogue of the space L p . Sov. Math. 23(11), 71–79 (1979) · Zbl 0477.46052 |
| [10] | Trunov N.V.: The L p -spaces associating with weight on semi finite von Neumann algebras. (Russian) Konstr. Teor. Funkts. Funkts. Anal. 3, 88–93 (1981) · Zbl 0514.46044 |
| [11] | Trunov N.V.: To the theory of normal weights on von Neumann algebras. Sov. Math. 26(2), 72–84 (1982) · Zbl 0521.46056 |
| [12] | Yeadon F.J.: Non-commutative L p -spaces. Math. Proc. Cambridge Phil. Soc. 77(1), 91–102 (1975) · Zbl 0327.46068 · doi:10.1017/S0305004100049434 |
| [13] | Zakirov B.S.: Noncommutative Arens algebras. Uzb. Math. J. 1, 17–24 (1997) · Zbl 0901.46061 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
