We develop a general theory of order isomorphisms of operator intervals. In this way we unify and extend several known results, among others the famous Ludwig's description of ortho-order automorphisms of effect algebras and Molnár's characterization of bijective order preserving maps on bounded observables. Besides proving several new results, one of the main contributions of the paper is to provide self-contained proofs of several known theorems whose original proofs depend on various deep results from functional analysis, operator algebras, and geometry. At the end we will show the optimality of the obtained theorems using Löwner's theory of operator monotone functions.
COBISS.SI-ID: 18263641
Botelho, Jamison, and Molnár , and Gehér and Šemrl have recently described the general form of surjective isometries of Grassmann spaces of all projections of a fixed finite rank on a Hilbert space $H$. As a straightforward consequence one can characterize surjective isometries of Grassmann spaces of projections of a fixed finite corank. In this paper we solve the remaining structural problem for surjective isometries on the set $P_\infty (H)$ of all projections of infinite rank and infinite corank when $H$ is separable. The proof technique is entirely different from the previous ones and is based on the study of geodesics in the Grassmannian $P_\infty (H)$. However, the same method gives an alternative proof in the case of finite rank projections.
COBISS.SI-ID: 18384473
Let $\mathbb{D}$ be a division ring, $n \geq 3$ an integer, and $P_n(\mathbb{D})$ the poset of all $n \times n$ idempotent matrices over $\mathbb{D}$ with the partial order defined by $P \leq Q$ if $PQ = QP = P$. Let $T \in M_n(\mathbb{D})$ be an invertible matrix and $\sigma : \mathbb{D} \to \mathbb{D}$ an endomorphism ($\tau : \mathbb{D} \to \mathbb{D}$ an anti-endomorphism). For any $P \in P_n(\mathbb{D})$ we denote by $P^\sigma$ ($P^\tau$) the idempotent matrix obtained from $P$ by applying $\sigma(\tau)$ entrywise. The map $\phi : P_n(\mathbb{D}) \to P_n(\mathbb{D})$ defined by $\phi(P) = TP^\sigma T^{- 1},\ P \in P_n(\mathbb{D})\ (\phi(P) = T ^t(P^\tau) T^{- 1},\ P \in P_n(\mathbb{D}))$ is an injective map preserving order in both directions. Every such map is called a standard map. It has been known before that if $\mathbb{D}$ is an EAS division ring, then every injective order preserving map $\phi : P_n(\mathbb{D}) \to P_n(\mathbb{D})$ is either standard or of a very special degenerate form. In this paper we use some ideas from geometry of algebraic homogeneous spaces and elementary field theory to give examples showing that the EAS assumption is indispensable. Then we define generalized standard maps and using them we describe the general form of injective order preservers on $P_n(\mathbb{D})$ for an arbitrary division ring $\mathbb{D}$. Our proof is shorter than the original one for the special case of EAS division rings. Under somewhat stronger assumptions we get...
COBISS.SI-ID: 18728281
Let $D$ be a division ring and let $\phi:M_n(D)\to M_n(D)$, $n\ge 2$, be a (not necessarily additive) map satisfying $\phi(A)\phi(B)=0$ whenever $AB=0$. We describe the form of $\phi$ under various assumptions on $\phi$, $n$ or $D$, and provide examples showing that these assumptions are necessary.
COBISS.SI-ID: 15279363
We prove that every continuous map acting on the four-dimensional Minkowski space and preserving light cones in one direction only is either a Poincaré similarity, that is, a product of a Lorentz transformation and a dilation, or it is of a very special degenerate form.
COBISS.SI-ID: 18656857