  In this paper we introduce the notion of slice regular right linear semigroup
in a quaternionic Banach space. It is an operatorial function which is slice
regular (a noncommutative counterpart of analyticity) and which satisfies a
noncommutative semigroup law characterizing the exponential function in an
infinite dimensional noncommutative setting. We prove that a right linear
operator semigroup in a quaternionic Banach space is slice regular if and only
if its generator is spherical sectorial. This result provides a connection
between the slice regularity and the noncommutative semigroups theory, and
characterizes those semigroups which can be represented by a noncommutative
Cauchy integral formula. All our results are generalized to Banach two-sided
modules having as a set of scalar any real associative *-algebra, Clifford R_n
algebras included.
