A set of polynomials in noncommuting variables is called locally linearly dependent if their evaluations at tuples of matrices are always linearly dependent. By a theorem of Camino, Helton, Skelton and Ye, a finite locally linearly dependent set of polynomials is linearly dependent. In this short note an alternative proof based on the theory of polynomial identities is given. The method of the proof yields generalizations to directional local linear dependence and evaluations in general algebras over fields of arbitrary characteristic. A main feature of the proof is that it makes it possible to deduce bounds on the size of the matrices where the (directional) local linear dependence needs to be tested in order to establish linear dependence.

COBISS.SI-ID: 16626521

Let $A$ be a unital $C^\ast$-algebra and $A''$ its second dual. By $\sigma(a)$ and $r(a)$ we denote the spectrum and the spectral radius of $a \in A$, respectively. The following two statements hold for arbitrary $a,b \in A$: (1) $\sigma(ac) \subseteq \sigma(bc) \cup \{0\}$ for every $c \in A$ if and only if there exists a central projection $z \in A''$ such that $a = zb$,(2) $r(ac) \le r(bc)$ for every $c \in A$ if and only if there exists a central element $z$ in $A''$ such that $a=zb$ and $\Vert z \Vert \le 1$.

COBISS.SI-ID: 16626777

Let $H$ be a Hilbert space and $E(H)$ the effect algebra on $H$, that is, $E(H)$ is the set of all self-adjoint operators $A \colon H \to H$ satisfying $0 \leqslant A \leqslant I$. The effect algebra can be equipped with several operations and relations that are important in mathematical foundations of quantum mechanics. Automorphisms with respect to these operations or relations are called symmetries. We present a new method that can be used to describe the general form of such maps. The main idea is to reduce this kind of problem to the study of adjacency-preserving maps. The efficiency of this approach is illustrated by reproving some known results as well as by obtaining some new theorems.

COBISS.SI-ID: 16756569