We consider certain functional identities on the matrix algebra $M_n$ that are defined similarly as the trace identities, except that the "coefficients" are arbitrary polynomials, not necessarily those expressible by the traces. The main issue is the question of whether such an identity is a consequence of the Cayley-Hamilton identity. We show that the answer is affirmative in several special cases, and, moreover, for every such an identity $P$ and every central polynomial $c$ with zero constant term there exists $m \in \mathbb{N}$ such that the affirmative answer holds for $c^mP$. In general, however, the answer is negative. We prove that there exist antisymmetric identities that do not follow from the Cayley-Hamilton identity, and give a complete description of a certain family of such identities.
COBISS.SI-ID: 17308505
Complete solutions of functional identities $\sum_{k\in K}F_k\left(\overline {x}_m^k\right)x_k = \sum_{l\in L}x_lG_l\left(\overline {x}_m^l\right)$ on the matrix algebra $M_n(\mathbb F)$ are given. The nonstandard parts of these solutions turn out to follow from the Cayley-Hamilton identity.
COBISS.SI-ID: 17540441
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 behavior 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.
COBISS.SI-ID: 17371993
Let $H$ be a Hilbert space and $E(H)$ the effect algebra on $H$. A bijective map $\phi \colon E(H) \to E(H)$ is called an ortho-order automorphism of $E(H)$ if for every $A, B \in E(H)$ we have $A \leqslant B \iff \phi(A) \leqslant (B)$ and $\phi (A^\perp) = \phi (A)^\perp$. The classical theorem of Ludwig states that every such $\phi$ is of the form $\phi(A) = UAU^\ast$, $A \in E(H)$, for some unitary or antiunitary operator $U$. It is also known that each bijective map on $E(H)$ preserving order and coexistency in both directions is of the same form. Can we improve these two theorems by relaxing the bijectivity assumption and/or replacing the above preserving properties by the weaker assumptions of preserving above relations in one direction only and still get the same conclusion? For both characterizations of automorphisms of effect algebras we will prove the optimal versions and give counterexamples showing the optimality of the obtained results.
COBISS.SI-ID: 17371737
We investigate constraints on embeddings of a non-orientable surface in a $4$-manifold with the homology of $M \times I$, where $M$ is a rational homology $3$-sphere. The constraints take the form of inequalities involving the genus and normal Euler class of the surface, and either the Ozsváth-Sazbó $d$-invariants or Atiyah-Singer $\rho$-invariants of $M$. One consequence is that the minimal genus of a smoothly embedded surface in $L(2p,q) \times I$ is the same as the minimal genus of a surface in $L(2p,q)$. We also consider embeddings of non-orientable surfaces in closed $4$-manifolds.
COBISS.SI-ID: 17557337