D’Alembert’s $\mu$-Matrix Functional Equation on Groups with an Anti-endomorphism

Abstract

Let G be a group and let $\mathcal {M}_{2}(\mathbb {C})$ be the algebra of all complex $2\times 2$ matrices. Using linear algebra, we study the solutions $H:G\rightarrow \mathcal {M}_{2}(\mathbb {C})$ of d’Alembert’s $\mu$-matrix functional equation:

$$\begin{aligned} H(xy)+\mu (y)H(x\psi (y))=2H(x)H(y), \quad x,y\in G, \end{aligned}$$

where $\psi :G\rightarrow G$ is an anti-endomorphism that need not be involutive and $\mu :G\rightarrow \mathbb {C}$ is a multiplicative function, such that $\mu (x\psi (x))=1$ for all $x\in G$.