Projective (or inverse) limit of C*-algebras

Question

(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)

Answer 1

The category of $C^*$-algebras is complete. The limit of a diagram $(A_i,\lVert - \rVert)_{i \in I}$ of $C^*$-algebras has as underlying $*$-algebra the $*$-subalgebra of the $*$-algebra $\prod_{i \in I} A_i$ whose elements $x=(x_i)_{i \in I}$ are subject to two conditions: First, the usual matching condition: For edges $i \to j$ the map $A_i \to A_j$ should map $x_i$ to $x_j$. Second, a boundedness condition: The set $\{\lVert x_i \rVert : i \in I\}$ is bounded. We then define $\lVert x \rVert := \sup_{i \in I} \lVert x_i \rVert$. It is straight forward to check that this is a $C^*$-norm. The only nontrivial fact is that the norm is complete. But you can simply copy the usual proof that $\ell^{\infty}$ is complete. It is also easy to check the universal property, using the fact that $*$-homomorphisms between $C^*$-algebras are of norm $\leq 1$.

Answer 2

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.