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.

For a multiplier pair ( X , Y ) \left ( X,Y\right ) we study the closed densely defined operators T T on X X that commute with all of the multiplications by right multipliers in X X . We apply our general results to special cases involving H p H^{p} , completions of L ∞ [ 0 , 1 ] L^{\infty }\left [ 0,1\right ] with respect to certain norms, and the completion of a I I 1 II_{1} factor von Neumann algebra with respect to a unitarily invariant norm, where we show that each such T T is a “left multiplication”. However, we give an example of a closed densely defined operator on the Bergman space that commutes with all multiplications by H ∞ H^{\infty } -functions but is not a multiplication operator.

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.

Let be a nest in an arbitrary factor von Neumann algebra . In this paper, it is proved that a linear local left (right) centralizer from the nest subalgebra into is a left (right) centralizer. Also, it is proved that a linear mapping from into itself is centralized at a given point if and only if is a centralizer.

In this paper, we will show that if the ordered pair (F, F*) is a Jordan elementary map of A × B. Then both F and F* are additive.

In this paper, we investigate a notion of spectrum $\sigma(f)$ for Banach algebra-valued holomorphic functions on $\mathbb{C}^{n}$. We prove that the resolvent $\sigma^{c}(f)$ is a disjoint union of domains of holomorphy when $\mathcal{B}$ is a $C^{\ast}$-algebra or is reflexive as a Banach space. Further, we study the topology of the resolvent via consideration of the $\mathcal{B}$-valued Maurer–Cartan type $1$-form $f(z)^{-1}\,df(z)$. As an example, we explicitly compute the spectrum of a linear function associated with the tuple of standard unitary generators in a free group factor von Neumann algebra.

We present a necessary and sufficient condition for a commutative subspace lattice (CSL) to have the property that every sublattice has the universal factorization property (UFP). Then we derive a sufficient condition for a CSL to have the UFP which applies to all known examples of CSL’s with the UFP. (The preceding results are, in fact, obtained for CSL- subalgebras of factor von Neumann algebras.) Finally, we give an example of a CSL which has the UFP, even though it is not a direct sum of a complemented lattice and a CSL with the property that every linearly ordered sublattice is countable.