## Hilbert C*-modules

An increasingly prominent tool in operator theory is the Hilbert C*-module, which are (loosely speaking) Hilbert spaces where the inner product takes values in a C*-algebra. The next level of generalization is that of Hilbert modules over locally C*-algebras (we briefly mentioned locally C*-algebras in this post), and much of the following theory extends to this setting as well.
Here I give the definition of a Hilbert C*-module and collect some of it’s properties, mostly as a reference for personal use. I will likely update this post with new material later on, hopefully without making it too bloated. The theory is now well developed in the literature so the proofs will kept to a bare minimum. For references I will mostly use [1] and [2].