Analog of Compact Operators

Question

This is kind of vague question, but I'll try to make it more precise.

$T$ is a compact operator on a Hilbert Space, $H$, if $\overline{T(D)}$ is compact in $H$, where of course, $D$ is the closed unit ball in $H$. So we use the underlying Hilbert space in order to define these operators. Now $B(H)$ is a $C^*$-algebra. Can we abstractly define/characterize compact operators only in terms of $C^*$-notions, without appealing to the underlying Hilbert space? Stated otherwise, is it possible to define, say "compact" elements in a $C^*$-algebra $A$, which agree with the usual notion if we take $A=B(H)$?

Answer 1

I am not familiar with such a concept, but you could call a projection $p$ in a $C^*$-algebra $A$ finite if $pAp$ is finite-dimensional; and then consider the closed ideal generated by the finite projections.

I don't know if this is useful, however.

Answer 2

(With the disclaimer that I am not an operator algebraist, so there may be technical details that are not quite right.)

There is a notion of "compact element" for a semifinite von Neumann algebra, which is roughly speaking one on which there is a faithful tracial weight whose ideal of definition is ultraweakly dense (at least in the case of von Neumann algebras whose predual is separable). The idea is similar to that described by Harald; you call a projection in the von Neumann algebra finite if its trace is a finite number, and then you define the ideal of compact elements to be the norm-closed ideal generated by the finite projections.

When your von Neumann algebra is $B(H)$, with the trace being the usual one, then a projection is finite in the sense above if and only if it has finite rank, and so one recovers $K(H)$.