论文标题
可分解的傅立叶乘数和操作员的敏感性表征
Decomposable Fourier Multipliers and an Operator-Algebraic Characterization of Amenability
论文作者
论文摘要
We study the algebra $\mathfrak{M}^{\infty,\mathrm{dec}}(G)$ of decomposable Fourier multipliers on the group von Neumann algebra $\mathrm{VN}(G)$ of a locally compact group $G$, and its relation to the Fourier-Stieltjes algebra $ \ mathrm {b}（g）$。对于离散组，我们证明这两个代数同时均匀。相比之下，我们表明身份$ \ mathfrak {m}^{\ infty，\ mathrm {dec}}（g）= \ mathrm {b}（g）$在各种非差异组的类别中失败，并且在第二个可容纳的单位组中，内在的单位群，内在的良好性可确保等值。我们的方法依赖于从$ \ mathrm {vn}（g）$上的完全有限的弱*连续运算符的空间中保留了完全积极性的承包预测的存在。我们表明，这种预测存在于内部的情况下。作为应用程序，我们获得了一种新的操作员 - 对敏感性的代数表征。我们还研究了非共同傅立叶乘数在非交换性$ \ mathrm {l}^p $ -spaces $ \ mathrm {l}^p（\ mathrm {vn}（g））$上的类似问题。使用谎言组理论和希尔伯特第五个问题的解决方案的结果，我们证明了第二个可估计的单模型有限维差不多的当地紧凑型组，允许兼容的预测在$ p = 1 $和$ p = \ iffty $。这些结果揭示了谐波分析，操作员代数和局部紧凑型组的几何形状之间的新结构联系。
We study the algebra $\mathfrak{M}^{\infty,\mathrm{dec}}(G)$ of decomposable Fourier multipliers on the group von Neumann algebra $\mathrm{VN}(G)$ of a locally compact group $G$, and its relation to the Fourier-Stieltjes algebra $\mathrm{B}(G)$. For discrete groups, we prove that these two algebras coincide isometrically. In contrast, we show that the identity $\mathfrak{M}^{\infty,\mathrm{dec}}(G) = \mathrm{B}(G)$ fails for various classes of non-discrete groups, and that, among second-countable unimodular groups, inner amenability ensures the equality. Our approach relies on the existence of contractive projections preserving complete positivity from the space of completely bounded weak* continuous operators on $\mathrm{VN}(G)$ onto the subspace of completely bounded Fourier multipliers. We show that such projections exist in the inner amenable case. As an application, we obtain a new operator-algebraic characterization of amenability. We also investigate the analogous problem for the space of completely bounded Fourier multipliers on the noncommutative $\mathrm{L}^p$-spaces $\mathrm{L}^p(\mathrm{VN}(G))$, for $1 \leq p \leq \infty$. Using Lie group theory and results stemming from the solution to Hilbert's fifth problem, we prove that second-countable unimodular finite-dimensional amenable locally compact groups admit compatible projections at $p = 1$ and $p = \infty$. These results reveal new structural links between harmonic analysis, operator algebras, and the geometry of locally compact groups.
