Characterizations of Minimal Elements in a Non-commutative $$L_p$$ -Space

For \(1\le p<\infty \), let \(L_p({\mathcal {M}},\tau )\) be the non-commutative \(L_p\)-space associated with a von Neumann algebra \({\mathcal {M}}\), where \({\mathcal {M}}\) admits a normal semifinite faithful trace \(\tau \). Using the trace \(\tau \), Banach duality formula and Gâteaux derivative, this paper characterizes an element \(a\in L_p({\mathcal {M}},\tau )\) such that

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

where \({\mathcal {B}}_p\) is a closed linear subspace of \(L_p({\mathcal {M}},\tau )\) and \(\Vert \cdot \Vert _p\) is the norm on \(L_p({\mathcal {M}},\tau )\). Such an a is called \({\mathcal {B}}_p\)-minimal. In particular, minimal elements related to the finite-diagonal-block type closed linear subspaces

$$\begin{aligned} {\mathcal {B}}_p=\bigoplus \limits _{i=1}^{\infty } e_i {\mathcal {S}} e_i \end{aligned}$$

(converging with respect to \(\Vert \cdot \Vert _p\)) are considered, where \(\{e_i\}_{i=1}^{\infty }\) is a sequence of mutually orthogonal and \(\tau \)-finite projections in a \(\sigma \)-finite von Neumann algebra \({\mathcal {M}}\), and \({\mathcal {S}}\) is the set of elements in \({\mathcal {M}}\) with \(\tau \)-finite supports.