标题： 完全有界双模映射与谱合成 显示英文标题

标题： Completely bounded bimodule maps and spectral synthesis

摘要： 我们开始研究局部紧群 $G$ 的两个拷贝的傅里叶代数 $A(G)$ 的Haagerup张量积 $A(G)\otimes_{\rm h} A(G)$ 的完全有界乘子。 如果$E$是$G$的闭子集，我们令$E^{\sharp} = \{(s,t) : st\in E\}$并证明如果$E^{\sharp}$是$A(G)\otimes_{\rm h} A(G)$的谱合成集，则$E$是$A(G)$的局部谱合成集。 相反，我们证明如果 $E$ 是 $A(G)$的谱合成集，且 $G$是 Moore 群，则 $E^{\sharp}$是 $A(G)\otimes_{\rm h} A(G)$的谱合成集。 使用所有完全有界弱*连续$VN(G)'$-双模态映射空间与$A(G)\otimes_{\rm h} A(G)$的对偶的自然识别，我们证明，当$G$是弱可约时，这样的映射使$L^{\infty}(G)$的乘法代数不变，当且仅当其支撑包含在$G$的反对角线上。 显示英文摘要

摘要： We initiate the study of the completely bounded multipliers of the Haagerup tensor product $A(G)\otimes_{\rm h} A(G)$ of two copies of the Fourier algebra $A(G)$ of a locally compact group $G$. If $E$ is a closed subset of $G$ we let $E^{\sharp} = \{(s,t) : st\in E\}$ and show that if $E^{\sharp}$ is a set of spectral synthesis for $A(G)\otimes_{\rm h} A(G)$ then $E$ is a set of local spectral synthesis for $A(G)$. Conversely, we prove that if $E$ is a set of spectral synthesis for $A(G)$ and $G$ is a Moore group then $E^{\sharp}$ is a set of spectral synthesis for $A(G)\otimes_{\rm h} A(G)$. Using the natural identification of the space of all completely bounded weak* continuous $VN(G)'$-bimodule maps with the dual of $A(G)\otimes_{\rm h} A(G)$, we show that, in the case $G$ is weakly amenable, such a map leaves the multiplication algebra of $L^{\infty}(G)$ invariant if and only if its support is contained in the antidiagonal of $G$.