论文标题
$ l^p $ - 交叉产品的下限
Lower bounds in $L^p$-transference for crossed-products
论文作者
论文摘要
令$γ\ curvearrowrightω$为一种措施的动作，$ \ Mathcal {l}γ\ HookrightArrow l^\ infty（ω）\ rtimesγ$ von Neumann代数的自然包含在交叉产品中。当$μ（ω）= \ infty $时，我们有这种天然嵌入不具有跟踪性的含量，因此不会限制到相关的非交换$ l^p $ - 空格。然而，我们表明，当$ω$不变时，意味着$ l^p（\ nathcal {l}γ）$的等距嵌入到$ l^p（ω\rtimesγ）$的超能中，它交织了傅立叶型乘数，并且是$ \ mathcal {l Mathcal {l} l}γ$ -Bimodular。结果，我们获得了较低的转移结合\ [ \ 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\ rtimesγ）\ to l^p（ω\ rtimesγ）\ big \ |，\]，完整规范也是如此。 具有不变平均值的条件是相当限制的。因此，我们探讨了其他epariant嵌入$φ是否：\ Mathcal {l}γ\ to l^\ infty（ω）$产生更一般的转移结果。我们表明，每当$φ$完全正面，可正常（在诱导正常的信件的意义上），上面的转移证明在逐字化，并且以$ l^2 $ level的方式将傅立叶乘数交织在一起。尽管没有获得新的转移结果，但同性图的分类及其研究性可能对某些读者具有独立的兴趣。
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.
