Minimal elements related to a conditional expectation in a C*-algebra

Let A be a C*-algebra and B a C*-subalgebra of A such that there is a conditional expectation from A onto it. Using the property of positive modification, this paper characterizes an element a∈ A satisfying ‖a‖=inf{‖a+b‖:b∈B}.Such an a is called B-minimal. As an application of these results it is shown that both the unilateral shift and the backward shift are D(B(l²)) -minimal, where D(B(l²)) is the set of diagonal operators in B(l²) , and thus provides new examples of minimal operators which are neither hermitian nor compact.