Minimal hermitian compact operators related to a C*-subalgebra of K(H)

Let K(H) be the C*-algebra of compact operators on a separable Hilbert space H. This paper studies the properties of hermitian compact operators Y such that ‖Y‖≤‖Y+W‖for all W∈W(H), where W(H) is a C*-subalgebra of K(H). Such a Y is called minimal related to W(H). The necessary and sufficient conditions that are required for Y to be minimal related to W(H) are characterized. Moreover, a particular C*-subalgebra W(H) such that there is a quasi-conditional expectation E from K(H) onto it is considered, and several examples are provided.