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 $\tau$. Let $d$ be an injective positive measurable operator with respect to $(\mathcal{M}, \tau)$ 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 $1\le p_0<p_1\le\8$, $0<\theta<1$ and $\alpha_0\ge0, \alpha_1\ge0$ the interpolation equality $$(L_{p_0}(d^{\alpha_0}), L_{p_1}(d^{\alpha_1}))_\theta =L_{p}(d^{\alpha})$$ holds with equivalent norms, where $\frac1p=\frac{1-\theta}{p_0}+\frac{\theta}{p_1}$ and $\alpha=(1-\theta)\alpha_0+\theta\alpha_1$.