Triple derivations on von Neumann algebras

Pluta, Robert; Russo, Bernard

doi:10.4064/sm226-1-3

Cited by 5 publications

(6 citation statements)

References 11 publications

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“…Jordan weak amenability is deeply connected with the more general notion of ternary weak amenability (see [25]). More interesting results on ternary weak amenability were recently developed by R. Pluta and B. Russo in [34].…”

Section: Jordan Derivationsmentioning

confidence: 95%

Jordan weak amenability and orthogonal forms on JB$^*$-algebras

Jamjoom¹,

Peralta²,

Siddiqui³

2015

Banach J. Math. Anal.

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We prove the existence of a linear isometric correspondence between the Banach space of all symmetric orthogonal forms on a JB *algebra J and the Banach space of all purely Jordan generalized derivations from J into J * . We also establish the existence of a similar linear isometric correspondence between the Banach spaces of all antisymmetric orthogonal forms on J , and of all Lie Jordan derivations from J into J * .Making use of the weak amenability of every C * -algebra, U. Haagerup and N.J. Laustsen gave a simplified proof of Goldstein's theorem in [21]. In the third section of the just quoted paper, and more concretely, in the proof of [21, Proposition 3.5], the above mentioned authors pointed out that for every anti-symmetric form V on a C * -algebra A which is orthogonal on A sa , the mappingReciprocally, the weak amenability of A also implies that every derivation δ from A into A * is inner and hence of the form δ(a) = adj φ (a) = φa − aφ for a functional φ ∈ A * . In particular, the form V δ (a, b) = δ(a)(b) is antisymmetric and orthogonal.The above results are the starting point and motivation of the present note. In the setting of C * -algebras we shall complete the above picture showing that symmetric orthogonal forms on a C * -algebra A are in bijective correspondence with the purely Jordan generalized derivations from A into A * (see Section 2 for definitions). However, the main goal of this note is to explore the orthogonal forms on a JB * -algebra and the similarities and differences between the associative setting of C * -algebras and the wider class of JB * -algebras.In Section 2 we revisit the basic theory and results on Jordan modules and derivations from the associative derivations on C * -algebras to Jordan derivations on C * -algebras and JB * -algebras. The novelties presented in this section include a new study about generalized Jordan derivations from a JB * -algebra J into a Jordan Banach J -module in the line explored in [31], [1, §4], and [8, §3]. We recall that, given a Jordan Banach J -module X over a JB * -algebra, a generalized Jordan derivation from J into X is a linear mapping G : J → X for which there exists ξ ∈ X * * satisfyingfor every a, b in J . We show how the results on automatic continuity of Jordan derivations from a JB * -algebra J into itself or into its dual, established by S. Hejazian, A. Niknam [23] and B. Russo and the second author of this paper in [33], can be applied to prove that every generalized Jordan derivation from J into J or into J * is continuous (see Proposition 2.1).Section 3 contains the main results of the paper. In Proposition 3.8 we prove that for every generalized Jordan derivation G : J → J * , where J is a JB * -algebra, the form V G : J × J → C, V G (a, b) = G(a)(b) is orthogonal on the whole J . We introduce the two new subclasses of purely Jordan generalized derivations and Lie Jordan derivations. A generalized derivation G : J → J * is said to be a purely Jordan generalized derivation if G(a)(b) = G(b)(a), for every a, b ∈ J ; while a Li...

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Section: Jordan Derivationsmentioning

confidence: 95%

Jordan weak amenability and orthogonal forms on JB$^*$-algebras

Jamjoom¹,

Peralta²,

Siddiqui³

2015

Banach J. Math. Anal.

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“…The following theorem provides some significant infinite dimensional examples of Lie algebras in which every derivation is inner. Its proof is in the spirit of [28]. Proof.…”

Section: Cohomology Of Lie Algebras With Involutionmentioning

confidence: 98%

“…By the theorem of Sinclair [33], S 0 is a derivation and by the theorems of Kadison and Sakai, [18,30], S 0 is an inner derivation, say S 0 (x) = ax − xa for some a ∈ V . By well known structure of the span of commutators in von Neumann algebras due to Pearcy-Topping, Halmos, Halpern, Fack-de la Harpe, and others (see [28] for the references), a = z + [c i , d i ], where c i , d i ∈ V and z belongs to the center of V . It follows that…”

Section: Cohomology Of Lie Algebras With Involutionmentioning

confidence: 99%

Cohomology of Jordan triples via Lie algebras

Chu¹,

Russo

2016

Topics in Functional Analysis And Algebra

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“…Two consequences of [117] are that finite dimensional von Neumann algebras and abelian von Neumann algebras have the property that every triple derivation into the predual is an inner triple derivation, analogous to the Haagerup result. This property was called normal ternary weak amenability in [205] where it is shown that it rarely holds in a general von Neumann algebra, but that it comes close. The main results of [205] are the following two theorems.…”

Section: Triple Derivations On Von Neumann Algebrasmentioning

confidence: 99%

“…This property was called normal ternary weak amenability in [205] where it is shown that it rarely holds in a general von Neumann algebra, but that it comes close. The main results of [205] are the following two theorems. The first one gives a cohomological characterization of finite factors and the second gives a "zero-one" law for factors.…”

Section: Triple Derivations On Von Neumann Algebrasmentioning

confidence: 99%

Triple derivations on von Neumann algebras

Cited by 5 publications

References 11 publications

Jordan weak amenability and orthogonal forms on JB$^*$-algebras

Jordan weak amenability and orthogonal forms on JB$^*$-algebras

Cohomology of Jordan triples via Lie algebras

Derivations and projections on Jordan triples: An introduction to nonassociative algebra, continuous cohomology, and quantum functional analysis

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