Murphy's {\em Positive definite kernels and Hilbert C${}^*$--modules} reorganized

TL;DR

This paper revisits and clarifies key results on positive definite kernels and Hilbert C*-modules originally presented in a 1997 publication, aiming to improve understanding and correct historical details.

Contribution

It provides a detailed reorganization and clarification of foundational results on positive definite kernels and Hilbert C*-modules, highlighting overlooked connections and correcting historical inaccuracies.

Findings

Clarified the structure of positive definite kernels in Hilbert C*-modules

Revealed previously unnoticed connections in the theory

Corrected minor historical inaccuracies in the literature

Abstract

The paper the title refers to is that in {\em Proceedings of the Edinburgh Mathematical Society}, {\bf 40} (1997), 367-374. Taking it as an excuse we intend to realize a twofold purpose: to atomize that important result showing by the way connections which are out of favour and to rectify a tiny piece of history.