Let $M_n$ be the multiplicative semigroup of all $n\times n$ complex matrices, and let $U_n$ and $G{L}_n$ be the $n$–degree unitary group and general linear group over complex number field, respectively. We characterize group homomorphisms from $U_n$ to $G{L}_m$ when $n>m\ge 1$ or $n=m\ge 3$, and thereby determine multiplicative homomorphisms from $U_n$ to $M_m$ when $n>m\ge 1$ or $n=m\ge 3$. This generalize Hochwald’s result in [Lin. Alg. Appl.  212/213:339-351(1994)]: if $f:U_n\to M_n$ is a spectrum–preserving multiplicative homomorphism, then there exists a matrix $R$ in $G{L}_n$ such...

Let $G$ be a group and $H$ an abelian group. Let $J^{*}(G,H)$ be the set of solutions $f:G\to H$ of the Jensen functional equation $f\left(xy\right)+f\left(x{y}^{-1}\right)=2f\left(x\right)$ satisfying the condition $f\left(xyz\right)-f\left(xzy\right)=f\left(yz\right)-f\left(zy\right)$ for all $x,y,z\in G$. Let $Q^{*}(G,H)$ be the set of solutions $f:G\to H$ of the quadratic equation $f\left(xy\right)+f\left(x{y}^{-1}\right)=2f\left(x\right)+2f\left(y\right)$ satisfying the Kannappan condition $f\left(xyz\right)=f\left(xzy\right)$ for all $x,y,z\in G$. In this paper we determine solutions of the Whitehead equation on groups. We show that every solution $f:G\to H$ of the Whitehead equation is of the form $4f=2\varphi +2\psi$, where $2\varphi \in J^{*}(G,H)$ and $2\psi \in Q^{*}(G,H)$. Moreover, if $H$ has the additional property that $2h=0$ implies $h=0$ for all $h\in H$,...

A space $X$ is splittable over a space $Y$ (or splits over $Y$) if for every $A\subset X$ there exists a continuous map $f:X\to Y$ with $f^{-1}fA=A$. We prove that any $n$-dimensional polyhedron splits over ${𝐑}^{2n}$ but not necessarily over ${𝐑}^{2n-2}$. It is established that if a metrizable compact $X$ splits over ${𝐑}^n$, then $dimX\le n$. An example of $n$-dimensional compact space which does not split over ${𝐑}^{2n}$ is given.