A manifold Fueter-Sce phenomenon in one hypercomplex variable

Fueter's theorem states, in modern terms, that the Laplacian maps slice-regular quaternionic functions into Fueter-regular functions with axial symmetry. This phenomenon is also present in the Clifford setting, where both slice-monogenic functions and generalized partial-slice monogenic functions are mapped by the Laplacian into monogenic functions with axial symmetry. These results are due, respectively, to Sce and Qian and to Xu and Sabadini. The present work puts the Fueter-Sce phenomenon into context for the wider class of strongly $T$-regular functions. It shows that the phenomenon appears over general associative $*$-algebras. Moreover, the symmetry considered here is multi-axial in a sense introduced by Eelbode. Additionally, but more surprisingly, the phenomenon studied by Fueter, Sce, Xu and Sabadini turns out to be the last step in a multi-step process. A new phenomenon in one hypercomplex variable is therefore discovered.