We prove that a Banach space E has the compact range property (CRP) if and only if for any given C∗-algebra A, every absolutely summing operator from A into E is compact. Related results for p-summing operators (0 < p < 1) are also discussed as well as operators on non-commutative L-spaces and C∗-summing operators.

As is well known absolute convergence and unconditional convergence for series are equivalent only in finite dimensional Banach spaces. Replacing the classical notion of absolutely summing operators by the notion of 1 summing operators