Closed densely defined operators commuting with multiplications in a multiplier pair

Nonlinear maps preserving Lie products on factor von Neumann algebras

Let [Formula: see text] be a factor von Neumann algebra with dim[Formula: see text]. In this paper, we prove that a map [Formula: see text] satisfies [Formula: see text] for all [Formula: see text] if and only if there exists a real number [Formula: see text] such that [Formula: see text], where [Formula: see text] is an additive ∗-derivation and [Formula: see text].

Let A be a unital *-algebra. In this paper, under some mild conditions on A, it is shown that a map ? : A ? Ais a nonlinear mixed bi-skew Lie triple derivation if and only if ? is an additive*-derivation. As applications, nonlinear mixed bi-skew Lie triple derivations on prime *-algebras, von Neumann algebras with no central summands of type I1, factor von Neumann algebras and standard operator algebras are characterized.

Let [Formula: see text] be a surjective map between some operator algebras such that [Formula: see text] for all [Formula: see text], where [Formula: see text] defined by [Formula: see text] and [Formula: see text] is Jordan product, i.e. [Formula: see text]. In this paper, we determine the concrete form of map [Formula: see text] on some operator algebras. Such operator algebras include standard operator algebras, properly infinite von Neumann algebras and nest algebras. Particularly, if [Formula: see text] is a factor von Neumann algebra that satisfies [Formula: see text] for all [Formula: see text] and idempotents [Formula: see text] then there exists nonzero scalar [Formula: see text] with [Formula: see text] such that [Formula: see text] for all [Formula: see text]

Let 𝒜 and ℬ be two factor von Neumann algebras. In this article, we prove that a nonlinear bijective map Φ : 𝒜 → ℬ satisfies Φ(X* Y + YX*) =Φ(X)*Φ(Y) + Φ(Y)Φ(X)* (∀X, Y ∈ 𝒜), if and only if Φ is a *-ring isomorphism. In particular, if 𝒜 and ℬ are type I factors, then Φ is a unitary isomorphism or conjugate unitary isomorphism.

<abstract><p>In this article, we proved that each nonlinear higher anti-derivable mapping on generalized matrix algebras is automatically additive. As for its applications, we find a similar conclusion on triangular algebras, full matrix algebras, unital prime rings with a nontrivial idempotent, unital standard operator algebras and factor von Neumann algebras respectively.</p></abstract>

Let be a unital ∗‐algebra with the unit I and be the symmetric center of . Assume that contains a nontrivial projection P such that implies A = 0 and implies A = 0. In this paper, we proved that if a map satisfies [ φ ( A ), B ] ∗ = −[ φ ( B ), A ] ∗ for all , then there exists a map such that φ ( A ) = Z A + f ( A ) for all , where . As applications, we obtain the concrete form of nonadditive skew commuting maps on prime ∗‐algebra, factor von Neumann algebras and standard operator algebras.

Let be a unital prime ring with characteristic not 2 and containing a nontrivial idempotent. Assume that is a surjective map. It is shown that f is strong 3-commutativity preserving, that is, f satisfies for all , if and only if for all , where is an element in the extended centroid of with and is a map from into its extended centroid. Applications to prime C-algebras, factor von Neumann algebras, standard operator algebras and matrix algebras are also obtained .

In this paper, we prove that every multiplicative anti-derivation map of an arbitrary generalized [Formula: see text]-matrix ring is additive. Even more, as consequence, we get the same conclusion for the class of triangular [Formula: see text]-matrix rings, unital prime rings with a nontrivial idempotent, standard operator algebras and factor von Neumann algebras.

Let A be a unital ∗-algebra containing a nontrivial projection. In this paper, we show that every nonlinear mixed bi-skew Jordan-type higher derivation on A is an additive higher ∗-derivation. Moreover, we apply the above result to prime ∗-algebras, factor von Neumann algebras and standard operator algebras.

We provide an alternative proof for the extreme amenability of the unitary group of the hyperfinite II${}_1$-factor von Neumann algebra, endowed with the strong operator topology.

We study the relation between multiplication operators M g with closed range on L a p and M g with closed range on H ∞ . Let p>0. For 0 < α < 1 , suppose g ∈ B α . Then the following are equivalent. M g : L a p → L a p is bounded below. M g : H ∞ → H ∞ is bounded below. 0 $ ]]> lim inf z → ∂D | g ( z ) | > 0 .

In this paper we consider linear operators M ψ C ϕ D and M ψ DC ϕ acting between weighted Bergman spaces, where M ψ , C ϕ and D are multiplication, composition and differentation operators respectively.Our goal is to characterize those holomorphic self maps ϕ of D for which M ψ DC ϕ and M ψ C ϕ D acts boundedly and compactly between weighted Bergman spaces.

Multiplication operators on the Bergman spaces of polygons

In this note, we prove that an operator between reproducing kernel Hilbert spaces is a multiplication operator if and only if it leaves invariant zero sets. To be more precise, it is shown that an operator T between reproducing kernel Hilbert spaces is a multiplication operator if and only if (Tf)(z)=0 holds for all f and z satisfying f(z)=0. As possible applications, we deduce a general reflexivity result for multiplier algebras, and furthermore prove fully vector-valued generalizations of mulitplier lifting results of Beatrous and Burbea.