Convolutions for Berezin quantization and Berezin-Lieb inequalities

Concepts and results from quantum harmonic analysis, such as the convolution between functions and operators or between two operators, are identified as the appropriate setting for Berezin quantization and Berezin-Lieb inequalities. Based on this insight, we provide a rigorous approach to the 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 the Berezin quantization and Weyl quantization.