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arXiv 2009.03019math.OAmath.FA
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交叉积的$L^p$ -迁移的下界

Lower bounds in $L^p$-transference for crossed-products

Adrián M. González-Pérez

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摘要：

设$Γ\curvearrowright Ω$是一个保测度作用，$\mathcal{L} Γ\hookrightarrow L^\infty(Ω) \rtimes Γ$是群vonNeumann代数在交叉积中的自然包含.当$μ(Ω) = \infty$，我们有，这个自然嵌入是不保迹的，因此不会有界地扩展到相关的非交换$L^p$ -空间。然而，我们发现，当$Ω$有一个不变的平均值有一个等距嵌入$L^p(\mathcal{L} Γ)$到超幂的$L^p(Ω\rtimes Γ)$交织傅立叶乘法器，是$\mathcal{L} Γ$ -双模。作为结果，我们得到的下转移界\[ \big\| T_m: L^p(\mathcal{L} Γ) \to L^p(\mathcal{L} Γ) \big\| \leq \big\| (\mathrm{id} \rtimes T_m): L^p(Ω\rtimes Γ) \to L^p(Ω\rtimes Γ) \big\|, \]和相同的以下完整的规范。具有不变平均值的条件是相当严格的。因此，我们探讨是否其他同变嵌入$Φ: \mathcal{L} Γ\to L^\infty(Ω)$产生更一般的迁移结果。我们表明，转移证明上述工程逐字$Φ$是完全积极的，顺从（在这个意义上诱导一个顺从的对应）和交织傅立叶乘法器在$L^2$ -水平。虽然没有得到新的迁移结果，但等变映射的分类及其顺从性的研究可能会引起一些读者的独立兴趣。

英文摘要：

Let $Γ\curvearrowright Ω$ be a measure-preserving action and $\mathcal{L} Γ\hookrightarrow L^\infty(Ω) \rtimes Γ$ the natural inclusion of the group von Neumann algebra into the crossed product. When $μ(Ω) = \infty$, we have that this natural embedding is not trace-preserving and therefore does not extends boundedly to the associated noncommutative $L^p$-spaces. Nevertheless, we show that when $Ω$ has an invariant mean there is an isometric embedding of $L^p(\mathcal{L} Γ)$ into an ultrapower of $L^p(Ω\rtimes Γ)$ that intertwines Fourier multipliers and is $\mathcal{L} Γ$-bimodular. As a consequence we obtain the lower transference bound \[ \big\| T_m: L^p(\mathcal{L} Γ) \to L^p(\mathcal{L} Γ) \big\| \leq \big\| (\mathrm{id} \rtimes T_m): L^p(Ω\rtimes Γ) \to L^p(Ω\rtimes Γ) \big\|, \] and the same follows for complete norms. The condition of having an invariant mean is quite restrictive. Therefore, we explore whether other equivariant embeddings $Φ: \mathcal{L} Γ\to L^\infty(Ω)$ yield a more general transference result. We show that the transference proof above works verbatim whenever $Φ$ is completely positive, amenable (in the sense of inducing an amenable correspondence) and intertwines Fourier multipliers at the $L^2$-level. Although no new transference results are obtained, both the classification of equivariant maps and the study their amenability may be of independent interest to some readers.

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