Ber, A. F. Continuous derivations on \(\ast\)-algebras of \(\tau\)-measurable operators are inner. (English. Russian original) Zbl 1281.46059 Math. Notes 93, No. 5, 654-659 (2013); translation from Mat. Zametki 93, No. 5, 658-664 (2013). Summary: It is proved that every continuous derivation on the \(\ast\)-algebra \(S(\mathcal M,\tau)\) of all \(\tau\)-measurable operators affiliated with a von Neumann algebra \(\mathcal M\) is inner. For every properly infinite von Neumann algebra \(\mathcal M\), any derivation on the \(\ast\)-algebra \(S(\mathcal M,\tau)\) is inner. It is proved that every continuous derivation on the \(\ast\)-algebra \(S(\mathcal M,\tau)\) of all \(\tau\)-measurable operators affiliated with a von Neumann algebra \(\mathcal M\) is inner. For every properly infinite von Neumann algebra \(\mathcal M\), any derivation on the \(\ast\)-algebra \(S(\mathcal M,\tau)\) is inner. Cited in 3 Documents MSC: 46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras 46L51 Noncommutative measure and integration Keywords:von Neumann algebra; properly infinite; \(\tau\)-measurable operator; continuous derivation PDF BibTeX XML Cite \textit{A. F. Ber}, Math. Notes 93, No. 5, 654--659 (2013; Zbl 1281.46059); translation from Mat. Zametki 93, No. 5, 658--664 (2013) Full Text: DOI References: [1] S. Sakai, \(C^{*}\)-Algebras and \(W^{*}\)-Algebras, Ergeb. Math. Grenzgeb., 60, Springer-Verlag, New York, 1971 · Zbl 0219.46042 [2] D. Olesen, “Derivations of \(AW^*\)-algebras are inner”, Pacific J. Math., 53:2 (1974), 555 – 561 · Zbl 0298.46064 · doi:10.2140/pjm.1974.53.555 [3] А. Ф. Бер, В. И. Чилин, Ф. А. Сукочев, “Дифференцирования в коммутативных регулярных алгебрах”, Матем. заметки, 75:3 (2004), 453 – 454 · Zbl 1071.47036 · doi:10.1023/B:MATN.0000023321.64184.5a · mi.mathnet.ru [4] A. F. Ber, V. I. Chilin, F. A. Sukochev, “Non-trivial derivations on commutative regular algebras”, Extracta Math., 21:2 (2006), 107 – 147 · Zbl 1129.46056 · eudml:41854 [5] А. Г. Кусраев, “Автоморфизмы и дифференцирования в расширенной комплексной \(f\)-алгебре”, Сиб. матем. журн., 47:1 (2006), 97 – 107 · Zbl 1113.46043 · emis:journals/SMZ/2006/01/97.htm · eudml:53130 [6] S. Albeverio, Sh. A. Ayupov, K. K. Kudaibergenov, “Structure of derivations on various algebras of measurable operators for type I von Neumann algebras”, J. Funct. Anal., 256:9 (2009), 2917 – 2943 · Zbl 1175.46054 · doi:10.1016/j.jfa.2008.11.003 [7] A. F. Ber, B. de Pagter, F. A. Sukochev, “Derivations in algebras of operator-valued functions”, J. Operator Theory, 66:2 (2011), 261 – 300 · Zbl 1247.47011 [8] А. Ф. Бер, “Непрерывность дифференцирований на собственно бесконечных \(*\)-алгебрах \(\tau\)-измеримых операторов”, Матем. заметки, 90:5 (2011), 776 – 780 · doi:10.4213/mzm9268 · mi.mathnet.ru [9] R. V. Kadison, J. R. Ringrose, Fundamentals of the Theory of Operator Algebras. Vol. II. Advanced Theory, Pure Appl. Math., 100, Academic Press, Orlando, FL, 1986 · Zbl 0601.46054 [10] М. А. Муратов, В. И. Чилин, Алгебры измеримых и локально измеримых операторов, Працi Iн-ту матем. НАН Украi\"ни, 69, Iн-т матем. НАН Украi\"ни, Киi\"в, 2007 · Zbl 1199.47002 [11] А. Ф. Бер, Б. де Пагтер, Ф. А. Сукочев, “Некоторые замечания о дифференцированиях в алгебрах измеримых операторов”, Матем. заметки, 87:4 (2010), 502 – 513 · Zbl 1242.46081 · doi:10.1134/S0001434610030259 · mi.mathnet.ru [12] J. Dixmier, Les alge\?bres d/opeŕateurs dans l/espace hilbertien (alge\?bres de von Neumann), Cahiers scientifiques, XXV, 2e revue et augmentee, Gauthier – Villars, Paris, 1969 · Zbl 0175.43801 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.