D'Alembert's functional equations on monoids with an anti-endomorphism (Q2175152)

The paper studies a generalization of d'Alembert's classic functional equation $g(x+y) + g(x-y) = 2g(x)g(y)$ for $x,y \in \mathbb{R}$. Here $g:\mathbb{R} \to \mathbb{C}$ is the unknown function. Let $M$ be a topological monoid, $\psi: M \to M$ a continuous map such that $\psi(xy) = \psi(y)\psi(x)$ for all $x,y \in M$, and $\mu:M \to \mathbb{C}$ a continuous, multiplicative function such that $\mu(x\psi(x)) = 1$ for all $x \in M$ ($\mu$ can be $1$). The main result of the paper is that the continuous solutions $g:M \to \mathbb{C}$ of the functional equation \[ g(xy) + \mu(y)g(x\psi(y)) = 2g(x)g(y) \text{ for all } x,y \in M, \] are the following ($\chi :M \to \mathbb{C}$ denotes continuous, multiplicative functions): \begin{itemize} \item[(i)] $g = \chi/2$, where $\chi \circ \psi = 0$. \item[(ii)] $g = (\chi + \mu\, \chi \circ \psi)/2$, where $\chi \circ \psi^2 = \chi$. \item[(iii)] $g = (\operatorname{tr} \pi)/2$, where $\pi$ is a continuous, irreducible representation of $M$ on $\mathbb{C}^2$ such that $\mu\, \pi \circ \psi= \text{adj} \circ \pi$. Here $\text{adj}$ is the adjugate map of $\mathbb{C}^2$ and \(\operatorname{tr}\) denotes the trace of the operator. \end{itemize} The main novelty of the paper is that $\psi$ is not necessarily an involution as in previous works like [\textit{T. M. K. Davison}, Publ. Math. 75, No. 1--2, 41--66 (2009; Zbl 1212.39034); \textit{H. Stetkær}, Functional equations on groups. Hackensack, NJ: World Scientific (2013; Zbl 1298.39018), Ch.~9].