Derivations on algebras of unbounded operators

Inoue, Atsushi; Ota, Shôichi

doi:10.1090/S0002-9947-1980-0580903-X

Derivations on algebras of unbounded operators
HTML articles powered by AMS MathViewer

by Atsushi Inoue and Shôichi Ota PDF

Trans. Amer. Math. Soc. 261 (1980), 567-577 Request permission

Abstract:

This paper is a study of derivations on unbounded operator algebras in connection with those in operator algebras. In particular we study spatiality of derivations in several situations. We give the characterization of derivations on general *-algebras by using positive linear functionals. We also show that a derivation with some range-property on a left $E{W^\# }$-algebra induced by an unbounded Hilbert algebra is strongly implemented by an operator which belongs to an algebra of measurable operators.

References

D. Arnal and J. P. Jurzak, Topological aspects of algebras of unbounded operators, J. Functional Analysis 24 (1977), no. 4, 397–425. MR 0435874, DOI 10.1016/0022-1236(77)90066-0
Ola Bratteli and Derek W. Robinson, Unbounded derivations of $C^{\ast }$-algebras, Comm. Math. Phys. 42 (1975), 253–268. MR 377526
Ola Bratteli and Derek W. Robinson, Unbounded derivations of $C^{\ast }$-algebras. II, Comm. Math. Phys. 46 (1976), no. 1, 11–30. MR 390785
Ola Bratteli and Derek W. Robinson, Unbounded derivations of von Neumann algebras, Ann. Inst. H. Poincaré Sect. A (N.S.) 25 (1976), no. 2, 139–164. MR 425632
Ola Bratteli, Unbounded derivations of operator algebras, Algèbres d’opérateurs et leurs applications en physique mathématique (Proc. Colloq., Marseille, 1977) Colloq. Internat. CNRS, vol. 274, CNRS, Paris, 1979, pp. 83–105 (English, with French summary). MR 560627

Les algèbres d’opérateurs dans l’espace Hilbertian

Paul R. Chernoff, Representations, automorphisms, and derivations of some operator algebras, J. Functional Analysis 12 (1973), 275–289. MR 0350442, DOI 10.1016/0022-1236(73)90080-3
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
Atsushi Inoue, On a class of unbounded operator algebras, Pacific J. Math. 65 (1976), no. 1, 77–95. MR 512382
Atsushi Inoue, Unbounded generalizations of standard von Neumann algebras, Rep. Mathematical Phys. 13 (1978), no. 1, 25–35. MR 482270, DOI 10.1016/0034-4877(78)90017-4
Atsushi Inoue and Ken Kuriyama, Topologies on unbounded operator algebras, Mem. Fac. Sci. Kyushu Univ. Ser. A 33 (1979), no. 2, 355–375. MR 548563, DOI 10.2206/kyushumfs.33.355
P. Kröger, Derivationen in $L^{+}(D)$, Wiss. Z. Karl-Marx-Univ. Leipzig Math.-Natur. Reihe 24 (1975), no. 5, 525–528. Beiträge zur Theorie nichtnormierbarer topologischer Algebren. MR 428062
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
I. Namioka and E. Asplund, A geometric proof of Ryll-Nardzewski’s fixed point theorem, Bull. Amer. Math. Soc. 73 (1967), 443–445. MR 209904, DOI 10.1090/S0002-9904-1967-11779-8
Robert T. Powers, Self-adjoint algebras of unbounded operators, Comm. Math. Phys. 21 (1971), 85–124. MR 283580
J. R. Ringrose, Automatic continuity of derivations of operator algebras, J. London Math. Soc. (2) 5 (1972), 432–438. MR 374927, DOI 10.1112/jlms/s2-5.3.432
Czesław Ryll-Nardzewski, On fixed points of semigroups of endomorphisms of linear spaces, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 55–61. MR 0215134
Shôichirô Sakai, Derivations of $W^{\ast }$-algebras, Ann. of Math. (2) 83 (1966), 273–279. MR 193528, DOI 10.2307/1970432

algebras and

algebras

Recent developments in the theory of unbounded derivations in

algebras

I. E. Segal, A non-commutative extension of abstract integration, Ann. of Math. (2) 57 (1953), 401–457. MR 54864, DOI 10.2307/1969729