A characterization of optimal entanglement witnesses

In this paper, we present a characterization of optimal entanglement witnesses in terms of positive maps and then provide a general method of checking optimality of entanglement witnesses. Applying it, we obtain new indecomposable optimal witnesses which have no spanning property. These also provide new examples which support a recent conjecture saying that the so-called structural physical approximations to optimal positive maps (optimal entanglement witnesses) give entanglement breaking maps (separable states).Comment: 1

Maps preserving peripheral spectrum of generalized products of operators

Let $\mathcal{A}_1$ and $\mathcal{A}_2$ be standard operator algebras on complex Banach spaces $X_1$ and $X_2$, respectively. For $k\geq2$, let $(i_1,...,i_m)$ be a sequence with terms chosen from $\{1,\ldots,k\}$, and assume that at least one of the terms in $(i_1,\ldots,i_m)$ appears exactly once. Define the generalized product $T_1* T_2*\cdots* T_k=T_{i_1}T_{i_2}\cdots T_{i_m}$ on elements in $\mathcal{A}_i$. Let $\Phi:\mathcal{A}_1\rightarrow\mathcal{A}_2$ be a map with the range containing all operators of rank at most two. We show that $\Phi$ satisfies that $\sigma_\pi(\Phi(A_1)*\cdots*\Phi(A_k))=\sigma_\pi(A_1*\cdots* A_k)$ for all $A_1,\ldots, A_k$, where $\sigma_\pi(A)$ stands for the peripheral spectrum of $A$, if and only if $\Phi$ is an isomorphism or an anti-isomorphism multiplied by an $m$th root of unity, and the latter case occurs only if the generalized product is quasi-semi Jordan. If $X_1=H$ and $X_2=K$ are complex Hilbert spaces, we characterize also maps preserving the peripheral spectrum of the skew generalized products, and prove that such maps are of the form $A\mapsto cUAU^*$ or $A\mapsto cUA^tU^*$, where $U\in\mathcal{B}(H,K)$ is a unitary operator, $c\in\{1,-1\}$.Comment: 17 page

Strong skew commutativity preserving maps on von Neumann algebras

Let ${\mathcal M}$ be a von Neumann algebra without central summands of type $I_1$. Assume that $\Phi:{\mathcal M}\rightarrow {\mathcal M}$ is a surjective map. It is shown that $\Phi$ is strong skew commutativity preserving (that is, satisfies $\Phi(A)\Phi(B)-\Phi(B)\Phi(A)^*=AB-BA^*$ for all $A,B\in{\mathcal M}$) if and only if there exists some self-adjoint element $Z$ in the center of ${\mathcal M}$ with $Z^2=I$ such that $\Phi(A)=ZA$ for all $A\in{\mathcal M}$. The strong skew commutativity preserving maps on prime involution rings and prime involution algebras are also characterized.Comment: 16 page