Related papers: Additive mappings preserving orthogonality between…

The operator space analogue of the {\em strong form} of the principle of local reflexivity is shown to hold for any von Neumann algebra predual, and thus for any $C^{*}$-algebraic dual. This is in striking contrast to the situation for…

We present several analogies between convex geometry and the theory of holomorphic line bundles on smooth projective varieties or K\"ahler manifolds. We study the relation between positive products and mixed volumes. We define and study a…

Let $\mathcal{A}$ and $\mathcal{B}$ be unital finite-dimensional complex algebras, each equipped with the unique Hausdorff vector topology. Denote by $\mathrm{Max}(\mathcal{A})=\{\mathcal{M}_1, \ldots, \mathcal{M}_p\}$ and…

Maps that preserve adjacency on the set of all invertible hermitian matrices over a finite field are characterized. It is shown that such maps form a group that is generated by the maps $A\mapsto PAP^{\ast}$, $A\mapsto A^{\sigma}$, and…

Let $A$ be a unital locally matrix algebra. Among the examples of such algebras are: (1) an infinite tensor product $\otimes M_{n_i}(\mathbb{F})$ of matrix algebras over a field $\mathbb{F}$, and (2) the Clifford algebra of a nondegenerate…

Let $H$ be a complex Hilbert space and let ${\mathcal P}(H)$ be the associated projective space (the set of rank-one projections). Suppose that $\dim H\ge 3$. We prove the following Wigner-type theorem: if $H$ is finite-dimensional, then…

Let $H$ be a complex Hilbert space and let ${\mathcal G}_{k}(H)$ be the Grassmannian formed by $k$-dimensional subspaces of $H$. Suppose that $\dim H>2k$ and $f$ is an orthogonality preserving injective transformation of ${\mathcal…

We show that it is consistent relative to a weakly compact cardinal that strong homology is additive and compactly supported within the class of locally compact separable metric spaces. This complements work of Marde\v{s}i\'{c} and Prasolov…

We establish two inequalities in real inner product spaces. The first is a multiplicative strengthening of the classical Hornich-Hlawka inequality: for all vectors $x, y, z$ in a real inner product space $H$ \[ \|x\|\,\|y\| +…

It is known that all $k$-homogeneous orthogonally additive polynomials $P$ over $C(K)$ are of the form $$ P(x)=\int_K x^k d\mu . $$ Thus $x\mapsto x^k$ factors all orthogonally additive polynomials through some linear form $\mu$. We show…

A standard combinatorial construction, due to Kontsevich, associates to any A-infinity algebra with an invariant inner product, an inhomogeneous class in the cohomology of the moduli spaces of Riemann surfaces with marked points. We…

We call a function $f: X\to Y$ $P$-preserving if, for every subspace $A \subset X$ with property $P$, its image $f(A)$ also has property $P$. Of course, all continuous maps are both compactness- and connectedness-preserving and the natural…

We show that tensor products of $k$ gradients of harmonic functions, with $k$ at least three, are dense in $C(\overline{\Omega})$, for any bounded domain $\Omega$ in dimension 3 or higher. The bulk of the argument consists in showing that…

For a positive integer $n$, let $M_n$ be the set of $n\times n$ complex matrices. Suppose $\|\cdot\|$ is the Ky Fan $k$-norm with $1 \le k \le mn$ or the Schatten $p$-norm with $1 \le p \le \infty$ ($p\ne 2$) on $M_{mn}$, where $m,n\ge 2$…

We consider the transformation $\ev$ which associates to any element in a K-algebra A a function on the the set of its K-points. This is the analogue of the fundamental Gelfand transform. Both $\ev$ and its dual $\ev^*$ are the maps from a…

Under suitable hypotheses on the ground field and on the matrix $M$, we discuss existence, uniqueness and properties of some additive decompositions of $M$ and of its image through a convergent series.

In the first part of this paper, we consider, in the context of an arbitrary hyperplane arrangement, the map between compactly supported cohomology to the usual cohomology of a local system. A formula (i.e., an explicit algebraic de Rham…

Let $A$ and $B$ be C*-algebras and $\varphi\colon A\to B$ be a $*$-homomorphism. We discuss the properties of the kernel and (co-)image of the induced map $\mathrm{K}_{0}(\varphi)\colon \mathrm{K}_{0}(A) \to \mathrm{K}_{0}(B)$ on the level…

We give necessary and sufficient conditions for a family of inner products in a finite-dimensional vector space $V$ over an arbitrary field $\mathbb{K}$ to have an orthogonal basis relative to all the inner products. Some applications to…

A linear mapping upon real n-dimensional space, where the dimension n is odd, has a real eigenvalue-eigenvector pair. The corresponding statement for complex vector spaces holds true for any dimension n, but should be easy to demonstrate…