The problem of zero divisors in convolution algebras of supersolvable Lie groups

TL;DR

This paper proves that convolution algebras on simply connected supersolvable Lie groups have no zero divisors, extending the Titchmarsh theorem and relating to the zero divisor conjecture in a topological context.

Contribution

It establishes a zero divisor-free property for convolution algebras on supersolvable Lie groups, a novel extension of classical results to this class of groups.

Findings

Convolution algebras on supersolvable Lie groups lack zero divisors.

Extension of Titchmarsh convolution theorem to these groups.

Connection to the topological zero divisor conjecture of Kaplansky.

Abstract

We prove a variant of the Titchmarsh convolution theorem for simply connected supersolvable Lie groups, namely we show that the convolution algebras of compactly supported continuous functions and compactly supported finite measures on such groups do not contain zero divisors. This can be also viewed as a topological version of the zero divisor conjecture of Kaplansky.