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Structure of derivations on various algebras of measurable operators for type I von Neumann algebras. (English) Zbl 1175.46054

The authors continue their investigations from [J. Funct.Anal.253, No.1, 287–302 (2007; Zbl 1138.46040) and Positivity 12, No.2, 375–386 (2008; Zbl 1155.46038)] about derivations on algebras of unbounded operators affiliated to von Neumann algebras. Given a von Neumann algebra \(M\) there is the algebra \(S(M)\) of measurable operators, the algebra \(LS(M)\) of locally measurable operators and, if there is a faithful normal semi-finite trace \(\tau\) on \(M\), the algebra \(S(M,\tau)\) of \(\tau\)-measurable operators.
In this paper, the case of type \(I\) is worked out in detail. Equipped with some structural results about these algebras, it is shown that a derivation on \(LS(M)\) is inner if and only if it is \(Z\)-linear, where \(Z\) is the center of \(M\). For every derivation on \(LS(M)\), there is a decomposition into an inner derivation and another one generated by a derivation on the center of \(LS(M)\). For type \(I_\infty\), every derivation on \(LS(M)\) is inner.
About subalgebras of \(LS(M)\) containing \(M\), it is shown that any \(Z\)-linear derivation is implemented by an element of \(LS(M)\). This applies to \(S(M)\) and \(S(M,\tau)\). With some additional work, it follows that, similar as above, for every derivation on \(S(M)\) (resp., \(S(M,\tau)\)) there is a decomposition into an inner derivation and another one generated by a derivation on the center of \(S(M)\) (resp., \(S(M,\tau)\)), and in the case of type \(I_\infty\), every derivation on \(S(M)\) (resp., \(S(M,\tau)\)) is inner. The consequences for the first Hochschild cohomology groups (i.e., quotient spaces of derivations by inner derivations) are summarized.

MSC:

46L51 Noncommutative measure and integration
46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras
47B47 Commutators, derivations, elementary operators, etc.
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[1] Albeverio, S.; Ayupov, Sh. A.; Kudaybergenov, K. K., Non-commutative Arens algebras and their derivations, J. Func. Anal., 253, 287-302 (2007) · Zbl 1138.46040
[2] Albeverio, S.; Ayupov, Sh. A.; Kudaybergenov, K. K., Derivations on the algebra of measurable operators affiliated with a type I von Neumann algebra, Siberian Adv. Math., 18, 86-94 (2008)
[3] Ber, A. F.; Chilin, V. I.; Sukochev, F. A., Non-trivial derivation on commutative regular algebras, Extracta Math., 21, 107-147 (2006) · Zbl 1129.46056
[4] Chilin, V. I.; Ganiev, I. G.; Kudaybergenov, K. K., The Gelfand-Naimark-Segal theorem for \(C^\ast \)-algebras over a ring measurable functions, Vladikavkaz. Math. J., 9, 16-22 (2007)
[5] Dales, H. G., Banach Algebras and Automatic Continuity (2000), Clarendon Press: Clarendon Press Oxford · Zbl 0981.46043
[6] Gutman, A. E., Banach bundles in the theory of lattice-normed spaces, (Order-Compatible Linear Operators. Order-Compatible Linear Operators, Trudy Ins. Mat., vol. 29 (1995), Sobolev Institute Press: Sobolev Institute Press Novosibirsk). (Order-Compatible Linear Operators. Order-Compatible Linear Operators, Trudy Ins. Mat., vol. 29 (1995), Sobolev Institute Press: Sobolev Institute Press Novosibirsk), Siberian Adv. Math.. (Order-Compatible Linear Operators. Order-Compatible Linear Operators, Trudy Ins. Mat., vol. 29 (1995), Sobolev Institute Press: Sobolev Institute Press Novosibirsk). (Order-Compatible Linear Operators. Order-Compatible Linear Operators, Trudy Ins. Mat., vol. 29 (1995), Sobolev Institute Press: Sobolev Institute Press Novosibirsk), Siberian Adv. Math., Siberian Adv. Math.. (Order-Compatible Linear Operators. Order-Compatible Linear Operators, Trudy Ins. Mat., vol. 29 (1995), Sobolev Institute Press: Sobolev Institute Press Novosibirsk). (Order-Compatible Linear Operators. Order-Compatible Linear Operators, Trudy Ins. Mat., vol. 29 (1995), Sobolev Institute Press: Sobolev Institute Press Novosibirsk), Siberian Adv. Math.. (Order-Compatible Linear Operators. Order-Compatible Linear Operators, Trudy Ins. Mat., vol. 29 (1995), Sobolev Institute Press: Sobolev Institute Press Novosibirsk). (Order-Compatible Linear Operators. Order-Compatible Linear Operators, Trudy Ins. Mat., vol. 29 (1995), Sobolev Institute Press: Sobolev Institute Press Novosibirsk), Siberian Adv. Math., Siberian Adv. Math., Siberian Adv. Math.. (Order-Compatible Linear Operators. Order-Compatible Linear Operators, Trudy Ins. Mat., vol. 29 (1995), Sobolev Institute Press: Sobolev Institute Press Novosibirsk). (Order-Compatible Linear Operators. Order-Compatible Linear Operators, Trudy Ins. Mat., vol. 29 (1995), Sobolev Institute Press: Sobolev Institute Press Novosibirsk), Siberian Adv. Math.. (Order-Compatible Linear Operators. Order-Compatible Linear Operators, Trudy Ins. Mat., vol. 29 (1995), Sobolev Institute Press: Sobolev Institute Press Novosibirsk). (Order-Compatible Linear Operators. Order-Compatible Linear Operators, Trudy Ins. Mat., vol. 29 (1995), Sobolev Institute Press: Sobolev Institute Press Novosibirsk), Siberian Adv. Math., Siberian Adv. Math.. (Order-Compatible Linear Operators. Order-Compatible Linear Operators, Trudy Ins. Mat., vol. 29 (1995), Sobolev Institute Press: Sobolev Institute Press Novosibirsk). (Order-Compatible Linear Operators. Order-Compatible Linear Operators, Trudy Ins. Mat., vol. 29 (1995), Sobolev Institute Press: Sobolev Institute Press Novosibirsk), Siberian Adv. Math.. (Order-Compatible Linear Operators. Order-Compatible Linear Operators, Trudy Ins. Mat., vol. 29 (1995), Sobolev Institute Press: Sobolev Institute Press Novosibirsk). (Order-Compatible Linear Operators. Order-Compatible Linear Operators, Trudy Ins. Mat., vol. 29 (1995), Sobolev Institute Press: Sobolev Institute Press Novosibirsk), Siberian Adv. Math., Siberian Adv. Math., Siberian Adv. Math., Siberian Adv. Math., 6, 35-102 (1996), (Part IV)
[7] Gutman, A. E.; Kusraev, A. G.; Kutateladze, S. S., The Wickstead problem, Sib. Electron. Math. Reports, 5, 293-333 (2008) · Zbl 1299.46004
[8] Kadison, R. V.; Ringrose, J. R., Cohomology of operator algebras. I. Type I von Neumann algebras, Acta Math., 126, 227-243 (1971) · Zbl 0209.44501
[9] Kaplansky, I., Modules over operator algebras, Amer. J. Math., 75, 839-859 (1953) · Zbl 0051.09101
[10] Kaplansky, I.; Kaplansky, I., Algebras of type I, Ann. of Math. (2), 56, 460-472 (1952) · Zbl 0047.35701
[11] Kusraev, A. G., Dominated Operators (2000), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0983.47025
[12] Kusraev, A. G., Cyclically compact operators in Banach spaces, Vladikavkaz. Math. J., 2, 10-23 (2000) · Zbl 1030.47052
[13] Kusraev, A. G., Automorphisms and derivations on a universally complete complex \(f\)-algebra, Siberian Math. J., 47, 77-85 (2006) · Zbl 1113.46043
[14] Muratov, M. A.; Chilin, V. I., *-Algebras of unbounded operators affiliated with a von Neumann algebra, J. Math. Sci., 140, 445-451 (2007) · Zbl 1098.47034
[15] Nelson, E., Notes on non-commutative integration, J. Funct. Anal., 15, 91-102 (1975)
[16] Saito, K., On the algebra of measurable operators for a general \(A W^*\)-algebra, Tohoku Math. J., 23, 525-534 (1971)
[17] Sakai, S., \(C^\ast \)-Algebras and \(W^\ast \)-Algebras (1971), Springer-Verlag · Zbl 0219.46042
[18] Segal, I., A non-commutative extension of abstract integration, Ann. of Math., 57, 401-457 (1953) · Zbl 0051.34201
[19] Takesaki, M., Theory of Operator Algebras, vol. 1 (1979), Springer: Springer New York
[20] Vladimirov, D. A., Boolean Algebras (1969), Nauka: Nauka Moscow, (in Russian) · Zbl 0189.05702
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