标题： 傅里叶乘子的绝对扩张 显示英文标题

标题： Absolute dilation of Fourier multipliers

摘要： 设${\mathcal M}$是一个带有正常半有限忠实（nsf）迹的冯诺依曼代数。 我们说一个算子$T :{\mathcal M}\to {\mathcal M}$是绝对可扩张的，如果存在另一个带有正常非有限迹的冯诺依曼代数$M$，一个单位正规迹保持的$\ast$-同态$J: {\mathcal M} \to M$，以及一个迹保持的$\ast$-自同构$U: M \to M$，使得$T^k = {\mathbb E}_J U^k J \quad \text{for all } k \geq 0,$，其中${\mathbb E}_J: M \to {\mathcal M}$是与$J$相关的条件期望。 对于离散的可解群$G$和一个诱导出单位完全正的傅里叶乘子$M_u: VN(G) \to VN(G)$的函数$u:G\to\mathbb{C}$，我们建立以下传递定理：算子$M_u$有一个绝对扩张当且仅当其相关的 Herz-Schur 乘子有。 从这个结果，我们推导出在这个设定下具有绝对扩张的傅里叶乘子的特征。 基于传递结果，我们构造了第一个已知的不具有绝对扩张的单位完全正的傅里叶乘子的例子。 这个例子出现在对称群${\mathcal S}_3$中，这是出现这种现象的最小群。 此外，我们证明了对于每个阿贝尔群$G$，每个傅里叶乘子总是具有一个绝对扩张。 显示英文摘要

摘要： Let ${\mathcal M}$ be a von Neumann algebra equipped with a normal semifinite faithful (nsf) trace. We say that an operator $T :{\mathcal M}\to {\mathcal M}$ is absolutely dilatable if there exist another von Neumann algebra $M$ with an nsf trace, a unital normal trace preserving $\ast$-homomorphism $J: {\mathcal M} \to M$, and a trace preserving $\ast$-automorphism $U: M \to M$ such that $T^k = {\mathbb E}_J U^k J \quad \text{for all } k \geq 0,$ where ${\mathbb E}_J: M \to {\mathcal M}$ is the conditional expectation associated with $J$. For a discrete amenable group $G$ and a function $u:G\to\mathbb{C}$ inducing a unital completely positive Fourier multiplier $M_u: VN(G) \to VN(G)$, we establish the following transference theorem: the operator $M_u$ admits an absolute dilation if and only if its associated Herz-Schur multiplier does. From this result, we deduce a characterization of Fourier multipliers with an absolute dilation in this setting. Building on the transference result, we construct the first known example of a unital completely positive Fourier multiplier that does not admit an absolute dilation. This example arises in the symmetric group ${\mathcal S}_3$, the smallest group where such a phenomenon occurs. Moreover, we show that for every abelian group $G$, every Fourier multiplier always admits an absolute dilation.