Nihonkai Math. J. 32 (2), 111-131, (2021)
KEYWORDS: Möbius gyrogroup, Möbius gyrovector space, 46C05, 20N05, 46C99, 46T99, 51M10, 83A05
We show that any open ball centered at the origin of a complex inner product space endowed with a slightly modified Möbius addition is a gyrocommutative gyrogroup. A Möbius scalar multiplication by complex numbers can be naturally introduced so that some of the axioms of real inner product gyrovector spaces are satisfied. Moreover, the modified Möbius addition on the open ball of a complex inner product space can be characterized by three fundamental requirements.