Category: Operator Theory

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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 – 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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