One-Slice Preserving Functions of a Quaternionic Variable

Regular functions on the skew-field of quaternions were introduced by
Gentili and Struppa some 10 years ago in order to give an analogue of
holomorphic functions in a non commutative setting. After a (short)
introduction, I will give a formula which allows us to simplify the
understanding of the *-product, which corresponds to the pointwise
product of holomorphic functions. The peculiar structure of quaternions,
foliated in copies of complex plane, drives naturally to consider the
classes of functions which preserve either one or all complex slices.
The main part of the talk will be devoted to characterize the functions
whose sum, *-product or conjugate preserve a slice. At the end, time permitting, I will address to the case of *-powers which shows an unexpected connection with a problem of algebraic geometry studied by Causa and Re. (Joint work with A. Altavilla)