Let $\mathbb{D}$ be a division ring with an involution $x \mapsto x^\star$, and $n \geq 2$ be an integer. Denote by $\mathcal{H}_n(\mathbb{D})$ the set of all $n$-by-$n$ Hermitian matrices with entries in $\mathbb{D}$, and by $\mathcal{A}\mathcal{H}_n(\mathbb{D})$ the set of all matrices $A-A^\star$ with $A \in \mathcal{M}_n(\mathbb{D})$. Here, we give a complete solution to the following problem: Determine all group homomorphisms from $\mathcal{H}_n(\mathbb{D})$ to $\mathbb{D}^n$ (respectively, from $\mathcal{A}\mathcal{H}_n(\mathbb{D})$ to $\mathbb{D}^n$ unless $(-)^\star$ is the identity) that take every matrix to a right linear combination of its columns. The solution to this problem was already known when $(-)^\star$ is the identity, and the novelty here lies in the generalization to arbitrary involutions, and in particular in the noncommutative case. These results are to be used in a subsequent article on subspaces of Hermitian matrices of bounded rank, and on large spaces of diagonalisable matrices.
Let $(\mathscr {E}, \langle \cdot , \cdot \rangle )$ be a Hilbert $C^*$-module over a $C^*$-algebra $\mathfrak {A}$. The space of adjointable operators on $\mathscr {E}$ is denoted by $\mathcal {L}(\mathscr {E})$. The sets of all states and pure states on $\mathfrak {A}$ are denoted by $\mathcal {S}(\mathfrak {A})$ and...
Sajjad Abedi, M. Moslehian· Mathematica Scandinavica· 0 citations
Let $\mathcal{H}$ be a separable infinite dimensional Hilbert space and $\mathcal{B}(\mathcal{H})$ the algebra of all bounded linear operators on $\mathcal{H}$. A subalgebra $\mathfrak{A}$ in $\mathcal{B}(\mathcal{H})$ has the left (resp.\ right) partial factorization property if for any invertible operator $S\in\mathc...
Let $\mathbb{k}$ be a finite field and $\mathcal{A}, \mathcal{B}$ be $\mathbb{k}$-linear $\mathsf{Ext}$-finite hereditary abelian categories. A theorem of Cramer asserts that, under suitable assumptions, a derived equivalence $\mathcal{D}^b(\mathcal{A}) \!\longrightarrow\! \mathcal{D}^b(\mathcal{B})$ between two such c...
Let $d_1,\ldots,d_r$ be pairwise relatively prime positive square-free integers, with $d_j\geq2$ for all $1\leq j\leq r$. Using elementary matrices and combinatorial mathematics, we give a complete classification of the irreducible $\mathbb{Z}_+$-modules over the domain $\mathbb{Z}[d^{\frac{1}{N_1}}_1,d^{\frac{1}{N_2}}...
Let $\Omega$ be a lattice in $\mathbb{C}$ with algebraic invariants and complex multiplication, let $\mathcal{E}$ be the elliptic curve associated with $\Omega$, and let $\wp$ be the Weierstrass function relative to $\Omega$. Set $ k:=\operatorname{End}(\mathcal{E}) \otimes_{\mathbb{Z}}\mathbb{Q}.$ We prove that if $t_...
Let $\mathpzc K$ be an arbitrary commutative ring containing an invertible element $\varepsilon$. Let ${\mathcal H}_{{\!\vartriangle\!}}(r)_{\mathpzc K}$ be the extended affine Hecke algebra of type $A$ with Hecke parameter $\varepsilon$, let $\Omega_{\mathpzc K}^{\otimes r}$ be the affine tensor space, and let ${\math...
Qiang Fu, Jun Hu· 0 citations
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