# Convolutions for Berezin quantization and Berezin-Lieb inequalities

## TL;DR

This paper explores the use of quantum harmonic analysis, specifically convolutions of functions and operators, to provide a rigorous and conceptual framework for Berezin quantization and Berezin-Lieb inequalities, linking them to Weyl quantization.

## Contribution

It offers a new, rigorous approach to generalized phase-space representations and Berezin-Lieb inequalities using quantum harmonic analysis, clarifying their connection to Weyl quantization.

## Key findings

- Provides a rigorous framework for Berezin quantization
- Reveals a conceptual link between Berezin and Weyl quantizations
- Offers a new perspective on phase-space representations

## Abstract

Notions and results from quantum harmonic analysis, such as the convolution between functions and operators or between two operators, is identified as the appropriate setting for Berezin quantization and Berezin-Lieb inequalities. Based on this insight we provide a rigorous approach to generalized phase-space representation introduced by Klauder-Skagerstam and their variants of Berezin-Lieb inequalities in this setting. Hence our presentation of the results of Klauder-Skagerstam gives a more conceptual framework, which yields as a byproduct an interesting perspective on the connection between Berezin quantization and Weyl quantization.