Characterizations of Minimal Elements in a Non-commutative \(L_p\)-Space

For \(1\le p<\infty \) $1\le p<\infty$ , let \(L_p({\mathcal {M}},\tau )\) $L_p(\mathcal{M},\tau )$ be the non-commutative \(L_p\) $L_p$ -space associated with a von Neumann algebra \({\mathcal {M}}\) $\mathcal{M}$ , where \({\mathcal {M}}\) $\mathcal{M}$ admits a normal semifinite faithful trace \(\tau \) $\tau$ . Using the trace \(\tau \) $\tau$ , Banach duality formula and Gâteaux derivative, this paper characterizes an element \(a\in L_p({\mathcal {M}},\tau )\) $a\in L_p(\mathcal{M},\tau )$ such that \(\begin{aligned} \Vert a\Vert _p=\inf \{\Vert a+b\Vert _p: b\in {\mathcal {B}}_p\}, \end{aligned}\) $$\begin{array}{c}\hfill {\Vert a\Vert }_p={inf\{\Vert a+b\Vert }_p:b\in {\mathcal{B}}_p\},\end{array}$$ where \({\mathcal {B}}_p\) ${\mathcal{B}}_p$ is a closed linear subspace of \(L_p({\mathcal {M}},\tau )\) $L_p(\mathcal{M},\tau )$ and \(\Vert \cdot \Vert _p\) ${\Vert ·\Vert }_p$ is the norm on \(L_p({\mathcal {M}},\tau )\) $L_p(\mathcal{M},\tau )$ . Such an a is called \({\mathcal {B}}_p\) ${\mathcal{B}}_p$ -minimal. In particular, minimal elements related to the finite-diagonal-block type closed linear subspaces \(\begin{aligned} {\mathcal {B}}_p=\bigoplus \limits _{i=1}^{\infty } e_i {\mathcal {S}} e_i \end{aligned}\) $$\begin{array}{c}\hfill {\mathcal{B}}_p=\underset{i=1}{\overset{\infty }{⨁}}{e}_i\mathcal{S}{e}_i\end{array}$$ (converging with respect to \(\Vert \cdot \Vert _p\) ${\Vert ·\Vert }_p$ ) are considered, where \(\{e_i\}_{i=1}^{\infty }\) ${\{e_i\}}_{i=1}^{\infty }$ is a sequence of mutually orthogonal and \(\tau \) $\tau$ -finite projections in a \(\sigma \) $\sigma$ -finite von Neumann algebra \({\mathcal {M}}\) $\mathcal{M}$ , and \({\mathcal {S}}\) $\mathcal{S}$ is the set of elements in \({\mathcal {M}}\) $\mathcal{M}$ with \(\tau \) $\tau$ -finite supports.