Complex interpolation of weighted noncommutative $L_p$-spaces

Abstract

Let $\mathcal{M}$ be a semifinite von Neumann algebra equipped with a semifinite normal faithful trace $τ$. Let $d$ be an injective positive measurable operator with respect to $(\mathcal{M}, τ)$ such that $d^{-1}$ is also measurable. Define $$L_p(d)={x\in L_0(\mathcal{M}) : dx+xd\in L_p(\mathcal{M})}\quad{and}\quad \|x\|_{L_p(d)}=\|dx+xd\|_p .$$ We show that for $1\le p_0<p_1\le\8$, $0<θ<1$ and $α_0\ge0, α_1\ge0$ the interpolation equality $$(L_{p_0}(d^{α_0}), L_{p_1}(d^{α_1}))_θ =L_{p}(d^α)$$ holds with equivalent norms, where $\frac1p=\frac{1-θ}{p_0}+\fracθ{p_1}$ and $α=(1-θ)α_0+θα_1$.