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

Abstract

Let ${\mathcal {A}}$ be a C*-algebra and ${\mathcal {B}}$ a C*-subalgebra of ${\mathcal {A}}$ such that there is a conditional expectation from ${\mathcal {A}}$ onto it. Using the property of positive modification, this paper characterizes an element $a\in {\mathcal {A}}$ satisfying

$$\begin{aligned} \Vert a\Vert =\inf \{\Vert a+b\Vert : b\in {\mathcal {B}}\}. \end{aligned}$$

Such an a is called ${\mathcal {B}}$-minimal. As an application of these results it is shown that both the unilateral shift and the backward shift are $D(B(l^2))$-minimal, where $D(B(l^2))$ is the set of diagonal operators in $B(l^2)$, and thus provides new examples of minimal operators which are neither hermitian nor compact.