# Lyapunov's Theorem for continuous frames

## TL;DR

This paper extends Lyapunov's theorem to continuous frames on non-atomic measure spaces, providing a new theoretical result that does not depend on the Kadison-Singer problem's recent solution.

## Contribution

It establishes a Lyapunov-type theorem for continuous frames without relying on the Kadison-Singer problem's recent proof.

## Key findings

- Proves a Lyapunov theorem for continuous frames on non-atomic measure spaces.
- Shows the result does not depend on the Kadison-Singer problem solution.

## Abstract

Akemann and Weaver (2014) have shown a remarkable extension of Weaver's $KS_r$ Conjecture (2004) in the form of approximate Lyapunov's theorem. This was made possible thanks to the breakthrough solution of the Kadison-Singer problem by Marcus, Spielman, and Srivastava (2015). In this paper we show a similar type of Lyapunov theorem for continuous frames on non-atomic measure spaces. In contrast with discrete frames, the proof of this result does not rely on the recent solution of the Kadison-Singer problem.