A variant of d'Alembert's and Wilson's functional equations for matrix valued functions (Q6617110)

When dealing with d'Alembert's equation one usually thinks of the standard form \N\[\Ng(x+y) + g(x-y) = 2\,g(x)\,g(y)\N\]\Non an abelian group with codomain \(\mathbb{C}\) or, in generalized versions, on monoids or semigroups, where \(+\) is replaced by the semigroup operation and \(-\) by an involution \(\sigma\), and the codomain is the field of complex numbers \(\mathbb{C}\). In the nonabelian case one has to take care that the operation need not be commutative. This gives \N\[\Ng(xy) + g({\sigma}(y)x) = 2\,g(y)\,g(x).\N\]\NHere we already took into account the notation of the authors in the case that the codomain is a noncommutative algebra of matrices.\N\NSimilarly, Wilson's equation in standard form is given by \N\[\Nf(x+y) + f(x-y) = 2\,f(x)\,g(y),\N\]\Nor, in the generalized form,\N\[\Nf(xy) + f({\sigma}(y)x) = 2\,g(y)\,f(x).\N\]\NThe results in the paper may be grouped in four short parts:\N\NIn the first part the authors enlarge old and recent results to the situation that the codomain of a solution \(g\) of d'Alembert's equation -- they call the function now \(\Phi\) -- is the matrix algebra \(M_n({\mathbb{C}})\), the codomain of solutions of Wilson's equation is \({\mathbb{C}}^n\) respect. \(M_n({\mathbb{C}})\).\N\NA second part deals with d'Alembert's equation in a special case. Let \(M\) be a monoid. Then \(M {\times} M\) is also a monoid with componentwise operation. On \(M \times M\) there is a natural involution, \((x,y) \mapsto (y,x)\). With this involution they study d'Alembert's equation in the matrix case with domain \(M \times M\) and codomain \(M_n({\mathbb{C}})\).\N\NFurthermore, a third part deals with Wilson's equation similarly to Part 2.\N\NFinally, in Part 4, they describe the solutions of d'Alembert's and Wilson's equation explicitely in the case that the domain is \(M \times M\) with \({\sigma} \colon (x,y) \mapsto (y,x)\) and codomain \({\mathbb{C}}^3\) respect. \(M_3({\mathbb{C}})\).\N\NA (not so short) list of references gives an overview on results for d'Alembert's and Wilson's equation in general.