Compact element in a C$^*$-algebra?

Question

Can we define a compact element of an arbitrary C$^*$-algebra $A$? For example, what are the compact elements of C$_0(X)$?

Answer 1

One can generalize some characteristic of compact operators and use that as a definition. But there is no general criterion.

For instance, $K(H)\subset B(H)$ is the C$^*$-algebra generated by the minimal projections. So you can say that $a\in A$ if it belongs to the closed span of the minimal projections. But in many situations you have no minimal projections, like in $C_0(\mathbb R)$. Here one can use another characterization of compact operators: they are the norm limits of linear combinations of finite-rank projections; equivalently, of finite-trace projections. So the analogy in $C_0(X)$ would be to take the norm-closure of the characteristic functions of compact subsets; but this would make every element in $C_0(X)$, "compact".