Maps preserving product XY-YX* on factor von Neumann algebras

Let A and B be two factor von Neumann algebras. For A, B is an element of A. define by [A, B](*) = AB - BA* the new product of A and B. In this paper, we prove that a nonlinear bijective map phi : A - \u3e B satisfies phi([A, B](*)) = [phi(A), phi(B)](*) for all A, B is an element of A if and only phi is a *-ring isomorphism. In particular, if the von Neumann algebras A and B are type I factors, then phi is a unitary isomorphism or a conjugate unitary isomorphism. (C) 2009 Elsevier Inc. All rights reserved

Hua's fundamental theorem of geometry of rectangular matrices over EAS division rings

The fundamental theorem of geometry of rectangular matrices describes the general form of bijective maps on the space of all $m\times n$ matrices over a division ring $\mathbb{D}$ which preserve adjacency in both directions. This result proved by Hua in the nineteen forties has been recently improved in several directions. One can study such maps without the bijectivity assumption or one can try to get the same conclusion under the weaker assumption that adjacency is preserved in one direction only. And the last possibility is to study maps acting between matrix spaces of different sizes. The optimal result would describe maps preserving adjacency in one direction only acting between spaces of rectangular matrices of different sizes in the absence of any regularity condition (injectivity or surjectivity). A division ring is said to be EAS if it is not isomorphic to any proper subring. It has been known before that it is possible to construct adjacency preserving maps with wild behaviour on matrices over division rings that are not EAS. For matrices over EAS division rings it has been recently proved that adjacency preserving maps acting between matrix spaces of different sizes satisfying a certain weak surjectivity condition are either degenerate or of the expected simple standard form. We will remove this weak surjectivity assumption, thus solving completely the long standing open problem of the optimal version of Hua's theorem.Comment: 31 page

Maps on matrix spaces

AbstractIt is well known that every automorphism of the full matrix algebra is inner. We give a short proof of this statement and discuss several extensions of this theorem including structural results for multiplicative maps on matrix algebras, characterizations of monotone and orthogonality preserving maps on idempotent matrices, some nonlinear preserver results, and some recent theorems concerning geometry of matrices. We show that all these topics are closely related and point out the connections with physics and geometry. Several open problems are posed

Domain Range Semigroups and Finite Representations

Relational semigroups with domain and range are a useful tool for modelling nondeterministic programs. We prove that the representation class of domain-range semigroups with demonic composition is not finitely axiomatisable. We extend the result for ordered domain algebras and show that any relation algebra reduct signature containing domain, range, converse, and composition, but no negation, meet, nor join has the finite representation property. That is any finite representable structure of such a signature is representable over a finite base. We survey the results in the area of the finite representation property

Determinant preserving maps on matrix algebras

AbstractLet Mn be the algebra of all n×n complex matrices. If φ:Mn→Mn is a surjective mapping satisfying det(A+λB)=det(φ(A)+λφ(B)), A,B∈Mn, λ∈C, then either φ is of the form φ(A)=MAN, A∈Mn, or φ is of the form φ(A)=MAtN, A∈Mn, where M,N∈Mn are nonsingular matrices with det(MN)=1

Wigner's theorem on Grassmann spaces

Wigner's celebrated theorem, which is particularly important in the mathematical foundations of quantum mechanics, states that every bijective transformation on the set of all rank-one projections of a complex Hilbert space which preserves the transition probability is induced by a unitary or an antiunitary operator. This vital theorem has been generalised in various ways by several scientists. In 2001, Molnár provided a natural generalisation, namely, he provided a characterisation of (not necessarily bijective) maps which act on the Grassmann space of all rank-n projections and leave the system of Jordan principal angles invariant (see [17] and [20]). In this paper we give a very natural joint generalisation of Wigner's and Molnár's theorems, namely, we prove a characterisation of all (not necessarily bijective) transformations on the Grassmann space which fix the quantity TrPQ (i.e. the sum of the squares of cosines of principal angles) for every pair of rank-n projections P and Q

Finite Representability of Semigroups with Demonic Refinement

Composition and demonic refinement $\sqsubseteq$ of binary relations are defined by \begin{align*} (x, y)\in (R;S)&\iff \exists z((x, z)\in R\wedge (z, y)\in S) R\sqsubseteq S&\iff (dom(S)\subseteq dom(R) \wedge R\restriction_{dom(S)}\subseteq S) \end{align*} where $dom(S)=\{x:\exists y (x, y)\in S\}$ and $R\restriction_{dom(S)}$ denotes the restriction of $R$ to pairs $(x, y)$ where $x\in dom(S)$. Demonic calculus was introduced to model the total correctness of non-deterministic programs and has been applied to program verification. We prove that the class $R(\sqsubseteq, ;)$ of abstract $(\leq, \circ)$ structures isomorphic to a set of binary relations ordered by demonic refinement with composition cannot be axiomatised by any finite set of first-order $(\leq, \circ)$ formulas. We provide a fairly simple, infinite, recursive axiomatisation that defines $R(\sqsubseteq, ;)$. We prove that a finite representable $(\leq, \circ)$ structure has a representation over a finite base. This appears to be the first example of a signature for binary relations with composition where the representation class is non-finitely axiomatisable, but where the finite representations for finite representable structures property holds

An extension of the Gleason–Kahane–Żelazko theorem: A possible approach to Kaplansky's problem

AbstractLet A and B be unital Banach algebras with B semisimple. Is every surjective unital linear invertibility preserving map φ:A→B a Jordan homomorphism? This is a famous open question, often called “Kaplansky's problem” in the literature. The Gleason–Kahane–Żelazko theorem gives an affirmative answer in the special case when B=C. We obtain an improvement of this theorem. Our result implies that in order to answer the question in the affirmative it is enough to show that φ(x2) and φ(x) commute for every x∈A. In this way we obtain a new proof of the Marcus–Purves theorem