设$\mathcal{A}$是因子von Neumann代数，$ϕ$是$A$上的$\ast$ -Jordan导子，即对每个$A,B \in \mathcal{A}$，$ϕ(A\diamond_{1} B) = ϕ(A)\diamond_{1} B + A\diamond_{1}ϕ( B)$，其中$A\diamond_{1} B = AB + BA^{\ast}$，则$ϕ$是加性$\ast$ -导子.
英文摘要：
Let $\mathcal{A}$ be a factor von Neumann algebra and $ϕ$ be the $\ast$-Jordan derivation on $A$, that is, for every $A,B \in \mathcal{A}$,
$ϕ(A\diamond_{1} B) = ϕ(A)\diamond_{1} B + A\diamond_{1}ϕ( B)$ where $A\diamond_{1} B = AB + BA^{\ast}$, then $ϕ$ is additive $\ast$-derivation.
We characterize completely bounded normal Jordan $*$-homomorphisms acting on von Neumann algebras. We also characterize completely positive isometries acting on noncommutative $\mathrm{L}^p$-spaces.
Let $\mathcal {A}$ be a unital $\ast$-algebra. For $A, B\in\mathcal{A}$, define by $[A, B]_{*}=AB-BA^{\ast}$ and $A\bullet B=AB+BA^{\ast}$ the new products of $A$ and $B$. In this paper, under some mild conditions on $\mathcal {A}$, it is shown that a map $Φ:\mathcal {A}\rightarrow \mathcal {A}$ satisfies $Φ([A\bullet B, C]_{*})=[Φ(A)\bullet B, C]_{*}+[A\bullet Φ(B), C]_{*}+[A\bullet B, Φ(C)]_{*}$ for all $A, B,C\in\mathcal {A}$ if and only if $Φ$ is an additive $*-$derivation. In particular, we apply the above result to prime $\ast$-algebras, von Neumann algebras with no central summands of t…
We study morphisms of the generalized quantum logic of tripotents in JBW*-triples and von Neumann algebras. Especially, we establish generalization of celebrated Dye's theorem on orthoisomorphisms between von Neumann lattices to this new context. We show one-to-one correspondence between maps on tripotents preserving orthogonality, orthogonal suprema, and reflection $u\to -u$, on one side, and their extensions to maps that are real linear on sets of elements with bounded range tripotents on the other side. In a more general description we show that quantum logic morphisms on tripotent structur…
Let $\mathcal{A}$ be a $*$-algebra and $\mathcal{M}$ be a $*$-$\mathcal A$-bimodule, we study the local properties of $*$-derivations and $*$-Jordan derivations from $\mathcal{A}$ into $\mathcal{M}$ under the following orthogonality conditions on elements in $\mathcal A$: $ab^*=0$, $ab^*+b^*a=0$ and $ab^*=b^*a=0$. We characterize the mappings on zero product determined algebras and zero Jordan product determined algebras. Moreover, we give some applications on $C^*$-algebras, group algebra, matrix algebras, algebras of locally measurable operators and von Neumann algebras.
Let $\mathcal{A}$ and $\mathcal{B}$ be two factor von Neumann algebras and $η$ be a non-zero complex number. A nonlinear bijective map $ϕ:\mathcal A\rightarrow\mathcal B$ has been demonstrated to satisfy $$ϕ([A,B]_{*}^η\diamond_η C)=[ϕ(A),ϕ(B)]_{*}^η\diamond_ηϕ(C)$$ for all $A,B,C\in\mathcal A.$ If $η=1,$ then $ϕ$ is a linear $*$-isomorphism, a conjugate linear $*$-isomorphism, the negative of a linear $*$-isomorphism, or the negative of a conjugate linear $*$-isomorphism. If $η\neq 1$ and satisfies $ϕ(I)=1,$ then $ϕ$ is either a linear $*$-isomorphism or a conjugate linear $*$-isomorphism.
The present paper is devoted to study of ring isomorphisms of $\ast$-subalgebras of Murray--von Neumann factors. Let $\cM,$ $\cN$ be von Neumann factors of type II$_1,$ and let $S(\cM),$ $S(\cN)$ be the $\ast$-algebras of all measurable operators affiliated with $\cM$ and $ \cN,$ respectively. Suppose that $\cA\subset S(\cM),$ $\cB\subset S(\cN)$ are their $\ast$-subalgebras such that $\cM\subset \cA,$ $\cN\subset \cB.$ We prove that for every ring isomorphism $Φ: \cA \to \cB$ there exist a positive invertible element $a \in \cB$ with $a^{-1}\in \cB$ and a real $\ast$-isomorphism $Ψ: \cM \to \…
Let $\mathcal{A}$ be a factor with dim$\mathcal{A}\geq2$. For $A, B\in\mathcal{A}$, define by $[A, B]_{*}=AB-BA^{\ast}$ and $A\bullet B=AB+BA^{\ast}$ the new products of $A$ and $B$. In this paper, it is proved that a map $Φ: \mathcal {A}\rightarrow \mathcal {A}$ satisfies $Φ([A, B]_{*}\bullet C)=[Φ(A), B]_{*}\bullet C+[A, Φ(B)]_{*}\bullet C+[A, B]_{*}\bullet Φ(C)$ for all $A, B,C\in\mathcal {A}$ if and only if $Φ$ is an additive $*-$derivation.
A reduction theory for operators in type $\rm{I}_{n}$ von Neumann algebras
本文研究了一类$\mathrm{I}_{n}$ von Neumann代数$\mathscr{A}$中算子的结构.受若尔当典范形定理的启发，我们的主要动机是弄清楚$\mathscr{A}$中算子$A$的结构与包含在相对交换子$\{A\}^{\prime} \cap\mathscr{A}$中的有界极大阿贝尔幂等元集在相似性下唯一的性质之间的关系.进一步，利用巴拿赫代数的$K$ -理论对这类算子进行了分类.一些观点和技巧来自冯·诺依曼的约化理论。
We investigate the situation when a normal positive linear unital map on a semifinite von Neumann algebra leaving the trace invariant does not change fixed quantum Renyi's entropy of the density of a normal state. It is also shown that such a map does not change the entropy of any density if and only if it is a Jordan *-isomorphism on the algebra.
Let $M_q(H_{\mathbb{R}})$ be the $q$-Gaussian von Neumann algebra associated with a separable infinite dimensional real Hilbert space $H_{\mathbb{R}}$ where $-1 < q < 1$. We show that $M_q(H_{\mathbb{R}}) \not \simeq M_0(H_{\mathbb{R}})$ for $-1 < q \not = 0 < 1$. The C$^\ast$-algebraic counterpart of this result was obtained recently in [BCKW22]. Using ideas of Ozawa we show that this non-isomorphism result also holds on the level of von Neumann algebras.
In this paper we study conditions under which, for an inclusion of finite von Neumann algebras $N \subseteq M$, we have the reduced amalgamated free product $\ast_N M$ is embeddable into $(R \bar{\otimes} N_1)^ω$ for some other finite von Neumann algebra $N_1$, where $R$ is the hyperfinite $\textrm{II}_1$ factor.
On the structure of isometries between noncommutative Lp spaces
证明了与von Neumann代数相关的非交换Lp空间之间的等距的一些结构结果。我们发现，当M_1具有Watanabe扩张性质的变体时，等距T：Lp（M_1）to Lp（M_2）（1 le p < infty，p not 2）可以正则地表示为某种简单形式.相反，如果M_1是“近似半有限的”（定义如下），这种形式总是定义等距。虽然这两个性质都没有被完全理解，我们表明，他们享有的所有半有限代数和超有限代数（没有I_2型的和项），加上其他。因此，该分类比Yeadon的半有限代数定理（及其最近在[JRS]中的改进）更强，并且证明使用了独立的技术。与此相关的是，我们研究的模块化理论的正投影从一个冯诺依曼代数到约旦图像的另一个冯诺依曼代数，并使用这样的投影构造新的Lp等距插值。一些补充的结果和问题也被提出。
Suppose $\mathcal{A}$ is a separable unital ASH C*-algebra, $\mathcal{R}$ is a sigma-finite II$_{\infty}$ factor von Neumann algebra, and $π,ρ:\mathcal{A}\rightarrow\mathcal{R}$ are unital $\ast$-homomorphisms such that, for every $a\in\mathcal{A}$, the range projections of $π\left( a\right) $ and $ρ\left( a\right) $ are Murray von Neuman equivalent in $\mathcal{R}% $. We prove that $π$ and $ρ$ are approximately unitarily equivalent modulo $\mathcal{K}_{\mathcal{R}}$, where $\mathcal{K}_{\mathcal{R}}$ is the norm closed ideal generated by the finite projections in $\mathcal{R}$. We also prove …
We prove that every surjective unital linear mapping which preserves invertible elements from a Banach algebra onto a C*-algebra carrying a faithful tracial state is a Jordan homomorphism thus generalising Aupetit's 1998 result for finite von Neumann algebras.
