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].

## Constructing new C*-algebras – Universal C*-algebras (1)

Universal C*-algebras are C*-algebras defined implicitly by relations and generators, much like group presentations. Contrary to the case with groups where the presentation can be constructed as a quotient of the free group, there is no analogous construction for C*-algebras, and we are not always guarantied the existence of a universal C*-algebras for any presentation.

This series of posts introduces the notion of universal C*-algebras. In this post I introduce the general construction of a universal C*-algebra, and a simple method for computing the the universal C*-algebra of a family of unitaries and (sufficiently nice) relations. In subsequent posts I hope to cover more general constructions with possibly non-unitary operators and more subtle relations.

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## Constructing new C*-algebras – Crossed product.

In this post I want to cover the concept of a crossed product C*-algebras. These are algebras which arise naturally as C*-algebras associated with dynamical systems and (as the name suggests) has many similarities with the semidirect of groups. It is a wide area of active research. To simplify things, unless stated otherwise, $\mathcal{A}$ will denote a unital C*-algebra, $G$ a discrete group and $H$ will always denote a complex Hilbert space. I will give references for the more general definitions when the C*-algebra is non-unital or the group is only assumed to be locally compact, as this would require the use of multiplier algebras and Haar measures/integrals making the exposition less approachable and the notation a lot less readable.

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## Constructing new C*-algebras – Injective and Projective limits

The injective (or direct) limit of C*-algebras is one way to construct new C*-algebras from directed system of C*-algebras (defined below), and is an essential tool in operator theory, so one may as well get acquainted with it. The projective limit (or inverse limit) is not as common it seems, but I will add it here for completeness. In this post I will try to give a definition of the construct by universal properties of colimits in the category of C*-algebras, but reducing the prerequisites from category theory to a bare minimum. The point is to highlight that similarities between direct limits of groups, rings, algebras etc., stems from the fact that they all solve the same universal problem in their respective categories, and to justify why some of these limits/colimits are preserved under certain transformations. Though the similarities may be evidenced, this is understandably (but also unfortunately) often not addressed in the classical references of operator theory, as a formal definition of a limit/colimit would be a significant digression.

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