This is an announcement for the paper "Complex interpolation of weighted noncommutative $L_p$-spaces" by Eric Ricard and Quanhua Xu.

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 $$L_p(d)=\left{x\in L_0(\mathcal{M});:; dx+xd\in L_p(\mathcal{M})\right}\quad\mbox{and}\quad |x|_{L_p(d)}=|dx+xd|_p,.$$ 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 $\alpha=(1-\theta)\alpha_0+\theta\alpha_1$.

Archive classification: math.FA math.OA

Mathematics Subject Classification: 46L50; 46M35; 47L15

Remarks: To appear in Houston J. Math

The source file(s), inter.tex: 37005 bytes, is(are) stored in gzipped form as 0906.5305.gz with size 12kb. The corresponding postcript file has gzipped size 90kb.

Submitted from: quanhua.xu@univ-fcomte.fr

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