A new approach to slice analysis via slice topology

TL;DR

This paper reviews and extends the theory of slice topology and slice analysis from quaternionic functions to more general algebraic settings, including real left alternative algebras and Euclidean spaces, establishing foundational properties and extension results.

Contribution

It introduces a generalized framework for slice analysis applicable to functions with values in real left alternative algebras and Euclidean spaces, expanding the scope of slice regularity theory.

Findings

Established new properties of slice regular functions in generalized algebraic contexts

Proved extension theorems for slice regular functions in these settings

Confirmed the validity of Taylor expansions for slice regular functions

Abstract

In this paper we summarize some known facts on slice topology in the quaternionic case, and we deepen some of them by proving new results and discussing some examples. We then show, following [18], how this setting allows us to generalize slice analysis to the general case of functions with values in a real left alternative algebra, which includes the case of slice monogenic functions with values in Clifford algebra. Moreover, we further extend slice analysis, in one and several variables, to functions with values in a Euclidean space of even dimension. In this framework, we study the domains of slice regularity, we prove some extension properties and the validity of a Taylor expansion for a slice regular function.