Logarithms Over a Real Associative Algebra

TL;DR

This paper explores the properties of logarithmic functions over real associative algebras, establishing conditions for the existence and uniqueness of logarithms similar to those over real and complex numbers.

Contribution

It extends the theory of calculus over associative algebras by characterizing logarithms and exponential functions in a broad class of algebras, including nil and Type-R or Type-C algebras.

Findings

Exponential function is injective in any multiplicative unital nil algebra.

Unique logarithm exists on the exponential's image in such algebras.

Logarithm behavior in these algebras closely resembles that over real and complex numbers.

Abstract

Extending the work of Freese and Cook, which develop the basic theory of calculus and power series over real associative algebras, we examine what can be said about the logarithmic functions over an algebra. In particular, we find that for any multiplicative unital nil algebra the exponential function is injective, and hence the algebra has a unique logarithm on the image of the exponential. We extend this result to show that for a large class of algebras, the logarithms behave incredibly similarly to the logarithms over the real and complex numbers depending on if they are "Type-R" or "Type-C" algebras.