Maps preserving product X*Y + YX* on factor von Neumann algebras

Let 𝒜 and ? be two factor von Neumann algebras. In this article, we prove that a nonlinear bijective map Φ?:?𝒜?→?? 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 ? are type I factors, then Φ is a unitary isomorphism or conjugate unitary isomorphism.     

Local 3-cocycles of von Neumann algebras

We show that every local 3-cocycle of a von Neumann algebra R into an arbitrary unital dual R-bimodule S is a 3-cocycle.     

A characterization of ∗-isomorphism on factor von Neumann algebras北大核心CSCD

Let H and K be complex Hilbert spaces. Let A and B be two factor von Neumann algebras acting on H and K respectively. A characterization of ∗-isomorphism between A and B is given. Let Φ: A → B be a bijection. If Φ(A∗B + B∗A) = Φ(A)∗Φ(B) + Φ(B)∗Φ(A) for all A,B ∈ A, then Φ is a linear or a conjugate linear ∗-isomorphism. ©, 2015, Chinese Academy of Sciences. All right reserved.     

On normal τ-measurable operators affiliated with semifinite von Neumann algebras

Let τ be a faithful normal semifinite trace on the von Neumann algebra M, 1 ≥ q > 0. The following generalizations of problems 163 and 139 from the book [1] to τ-measurable operators are obtained; it is established that: 1) each τ-compact q-hyponormal operator is normal; 2) if a τ-measurable operator A is normal and, for some natural number n, the operator A ⁿ is τ-compact, then the operator A is also τ-compact. It is proved that if a τ-measurable operator A is hyponormal and the operator A ² is τ-compact, then the operator A is also τ-compact. A new property of a nonincreasing rearrangement of the product of hyponormal and cohyponormal τ-measurable operators is established. For normal τ-measurable operators A and B, it is shown that the nonincreasing rearrangements of the operators AB and BA coincide. Applications of the results obtained to F-normed symmetric spaces on (M, τ) are considered.     

A note on Baskakov-Kantorovich type operators preserving e−x

In this paper, we give a generalization of the Baskakov-Kantorovich type operators that reproduce functions e₀ and e^−x. We discuss uniform convergence of this generalization by means of the modulus of continuity and establish quantitive asymptotic formula.     

von Neumann��s mean ergodic theorem on complete random inner product modules

We first prove two forms of von Neumann’s mean ergodic theorems under the framework of complete random inner product modules. As applications, we obtain two conditional mean ergodic convergence theorems for random isometric operators which are defined on L _ℱ ^p (ℰ, H) and generated by measure-preserving transformations on Ω, where H is a Hilbert space, L ^p (ℰ, H) (1 ⩽ p < ∞) the Banach space of equivalence classes of H-valued p-integrable random variables defined on a probability space (Ω, ℰ, P), F a sub σ-algebra of ℰ, and L _ℱ ^p (ℰ(E,H) the complete random normed module generated by L ^p (ℰ, H).     

Continuity and Schatten–von Neumann Properties for Localization Operators on Modulation Spaces

We use sharp convolution estimates for weighted Lebesgue and modulation spaces to obtain an extension of the celebrated Cordero-Gröchenig theorems on boundedness and Schatten–von Neumann properties of localization operators on modulation spaces. We also give a new proof of the Weyl connection based on the kernel theorem for Gelfand–Shilov spaces.     

On ＂Problems on von Neumann Algebras by R. Kadison, 1967＂

A brief summary of the development on Kadison‘s famous problems (1967) is given. A new set of problems in von Neumann algebras is listed.     

Notes on von Neumann–Jordan and James Constants for Absolute Norms on {\mathbb{R}^2}

Let ${\|\cdot\|_{\psi}}$ be the absolute norm on ${\mathbb{R}^2}$ corresponding to a convex function ${\psi}$ on [0, 1] and ${C_{\text{NJ}}(\|\cdot\|_{\psi})}$ its von Neumann–Jordan constant. It is known that ${\max \{M_1^2, M_2^2\} \leq C_{\text{NJ}}(\| \cdot \|_{\psi}) \leq M_1^2 M_2^2}$ , where ${M_1 = \max_{0 \leq t \leq 1} \psi(t)/ \psi_2(t)}$ , ${M_2 = \max_{0\leq t \leq 1} \psi_2(t)/ \psi(t)}$ and ${\psi_2}$ is the corresponding function to the ? ₂-norm. In this paper, we shall present a necessary and sufficient condition for the above right side inequality to attain equality. A corollary, which is valid for the complex case, will cover a couple of previous results. Similar results for the James constant will be presented.