TITLE: Projective (or inverse) limit of C*-algebras
QUESTION [5 upvotes]: (I think that the term "inverse limit" is used when the index set is directed)

To begin with, I'd like to know if projective limits of C*-algebras (in the category of C*-algebras) always exist, and if not, what are some sufficient conditions for their existence.

A rapid look at various articles in internet shows that people have been interested in projective limits of C*-algebras but within categories of topological algebras, e.g. Operator Algebra and its Applications: Vol. 1, David E. Evans, Masamichi Takesaki p.130-131

In that particular excerpt, it says that the usual construction (which they recall equation ($*$)) does not yield the projective limit. I would be really interested to know why it fails, what should be modified to get the limit. (With my limited experience of C*-algebras, there often seems to be difficulties with the choice of a norm)

(I would also appreciate help for the tags if the question is also of interest in other fields)

REPLY [3 votes]: The category of $C^*$-algebras is complete like Martin Brandenburg explained. However, given an inverse system of $C^*$-algebras, it is often more convenient to consider its limit in the category of all topological $*$-algebras. The limit of a diagram $(A_i,\lVert - \rVert)_{i \in I}$ of $C^*$-algebras is then the topological $*$-subalgebra of the topological $*$-algebra $\prod_{i \in I} A_i$ (in the product topology) whose elements $x=(x_i)_{i \in I}$ are subject only to the usual matching condition but not the boundedness condition. See the following paper:
http://jot.theta.ro/jot/archive/1988-019-001/1988-019-001-010.pdf
for more detail.