An element of a group is called strongly reversible or strongly real if it can be expressed as a product of two involutions. We provide necessary and sufficient conditions for an element of $\mathrm{SL}(n,\mathbb{C})$ to be a product of two involutions. In particular, we classify the strongly reversible conjugacy classes in $\mathrm{SL}(n,\mathbb{C})$.

Isoclinic subspaces have been studied for over a century. Quantum error correcting codes were recently shown to define a subclass of families of isoclinic subspaces. The Knill–Laflamme theorem is a seminal result in the theory of quantum error correction, a central topic in quantum information. We show there is a generalized version of the Knill–Laflamme result and conditions that applies to all families of isoclinic subspaces. In the case of quantum stabilizer codes, the expanded conditions are shown to capture logical operators. We apply the general conditions to give a new perspective on a classical subclass of isoclinic subspaces defined by the graphs of anti-commuting unitary operators. We show how the result applies to recently studied mutually unbiased quantum measurements (MUMs), and we give a new construction of such measurements motivated by the approach.

Let A be a semisimple, unital, and complex Banach algebra. It is well known and easy to prove that A is commutative if and only $e^xe^y=e^{x+y}$ for all $x,y\in A$. Elaborating on the spectral theory of commutativity developed by Aupetit, Zemánek, and Zemánek and Pták, we derive, in this paper, commutativity results via a spectral comparison of $e^xe^y$ and $e^{x+y}$.

Let ${{M}_{n}}$ be the algebra of all $n\,\times \,n$ matrices over $\mathbb{C}$ . We say that $A,B\in {{M}_{n}}$ quasi-commute if there exists a nonzero $\xi \,\in \,\mathbb{C}$ such that $AB\,=\,\xi BA$ . In the paper we classify bijective not necessarily linear maps $\Phi :{{M}_{n}}\to {{M}_{n}}$ which preserve quasi-commutativity in both directions.

The Kronecker modules, $\mathbb{V}\left( m,h,\alpha\right)$ , where $m$ is a positive integer, $h$ is a height function, and $\alpha $ is a $K$ -linear functional on the space $K(X)$ of rational functions in one variable $X$ over an algebraically closed field $K$ , are models for the family of all torsion-free rank-2 modules that are extensions of finite-dimensional rank-1 modules. Every such module comes with a regulating polynomial $f$ in $K(X)[Y]$ . When the endomorphism algebra of $\mathbb{V}\left( m,h,\alpha\right)$ is commutative and non-trivial, the regulator $f$ must be quadratic in $Y$ . If $f$ has one repeated root in $K(X)$ , the endomorphism algebra is the trivial extension $K\ltimes S$ for some vector space $S$ . If $f$ has distinct roots in $K(X)$ , then the endomorphisms form a structure that we call a bridge. These include the coordinate rings of some curves. Regardless of the number of roots in the regulator, those End $\mathbb{V}\left( m,h,\alpha\right)$ that are domains have zero radical. In addition, each semi-local End $\mathbb{V}\left( m,h,\alpha\right)$ must be either a trivial extension $K\ltimes S$ or the product $K\times K$ .

Purely simple Kronecker modules $M$ , built from an algebraically closed field $K$ , arise from a triplet $\left( m,\,h,\,\alpha \right)$ where $m$ is a positive integer, $h:\,K\,\cup \,\{\infty \}\,\to \,\{\infty ,\,0,\,1,\,2,\,3,\,.\,.\,.\,\}$ is a height function, and $\alpha$ is a $K$ -linear functional on the space $K\left( X \right)$ of rational functions in one variable $X$ . Every pair $\left( h,\,\alpha \right)$ comes with a polynomial $f$ in $K\left( X \right)\left[ Y\, \right]$ called the regulator. When the module $M$ admits non-trivial endomorphisms, $f$ must be linear or quadratic in $Y$ . In that case $M$ is purely simple if and only if $f$ is an irreducible quadratic. Then the $K$ -algebra End $M$ embeds in the quadratic function field ${K\left( X \right)\left[ Y\, \right]}/{(f\,)}\;$ . For some height functions $h$ of infinite support $I$ , the search for a functional $\alpha$ for which $\left( h,\,\alpha \right)$ has regulator $0$ comes down to having functions $\eta \,:\,I\,\to \,K$ such that no planar curve intersects the graph of $\eta$ on a cofinite subset. If $K$ has characterictic not $2$ , and the triplet $\left( m,\,h,\,\alpha \right)$ gives a purely-simple Kronecker module $M$ having non-trivial endomorphisms, then $h$ attains the value $\infty$ at least once on $\text{K}\,\cup \,\text{ }\!\!\{\!\!\text{ }\infty \text{ }\!\!\}\!\!\text{ }$ and $h$ is finite-valued at least twice on $\text{K}\,\cup \,\text{ }\!\!\{\!\!\text{ }\infty \text{ }\!\!\}\!\!\text{ }$ . Conversely all these $h$ form part of such triplets. The proof of this result hinges on the fact that a rational function $r$ is a perfect square in $K\left( X \right)$ if and only if $r$ is a perfect square in the completions of $K\left( X \right)$ with respect to all of its valuations.

Let T be a linear transformation acting on the space of n x n complex matrices. Let G(k) be the set of all hermitian matrices with k positive and n — k negative eigenvalues. Let T map some indefinite inertia class G(k) onto itself. We classify all such T. The possibilities are congruence, congruence followed by transposition, and, if n = 2k, it is possible that — T can be a congruence or a congruence followed by transposing. In other words, negation is an admissible transformation when n = 2k.