Möbius’s functional equation and Schur’s lemma with applications to the complex unit disk

Abstract

© 2016, Springer International Publishing. Möbius addition is defined on the complex open unit disk by (Formula presented.) and Möbius’s exponential equation takes the form L(a⊕Mb) = L(a) L(b) , where L is a complex-valued function defined on the complex unit disk. In the present article, we indicate how Möbius’s exponential equation is connected to Cauchy’s exponential equation. Möbius’s exponential equation arises when one determines the irreducible linear representations of the unit disk equipped with Möbius addition, considered as a nonassociative group-like structure. This suggests studying Schur’s lemma in a more general setting.