Cauchy-Riemann operators and local slice analysis over real alternative algebras

We prove some formulas relating Cauchy-Riemann operators defined on hypercomplex subspaces of an alternative *-algebra to a differential operator associated with the concept slice-regularity and spherical Dirac operator. These results in particular allow introduce definition locally slice-regular function open path for local slice analysis. Since factor corresponding Laplacian operators, proven let us also obtain several about harmonicity polyharmonicity properties functions.