On the ideal of compact operators on a Hilbert C⁎-module

This note studies certain properties of the ideal K(E) consisting of compact operators on E, where E is a Hilbert C⁎-module over a C⁎-algebra A. It is proved that this ideal has σp-unit, simpleness, separability, real rank zero, and stable rank one under certain conditions. An counterexample is given to illustrate that two Hilbert C⁎-modules may be not unitarily equivalent when their ideals of compact operators are isomorphic. Hereditary C⁎-subalgebras of such an ideal are also characterized.