# On spectral gaps of Markov maps

## TL;DR

This paper demonstrates that spectral gaps of Markov maps on noncommutative probability spaces extend across different Lp spaces, with the converse holding under factorizability, providing new insights even in classical settings.

## Contribution

It establishes the equivalence of spectral gaps across Lp spaces for Markov maps and introduces conditions for the converse, advancing understanding in noncommutative and classical probability.

## Key findings

- Spectral gap on L2 implies spectral gap on Lp for 1<p<∞.
- Converse holds if the Markov map is factorizable.
- Results are new for classical probability spaces.

## Abstract

It is shown that if a Markov map $T$ on a noncommutative probability space $\mathcal{M}$ has a spectral gap on $L_2(\mathcal{M})$, then it also has one on $L_p(\mathcal{M})$ for $1<p<\infty$. For fixed $p$, the converse also holds if $T$ is factorizable. These results are also new for classical probability spaces.