Noncommutative Fourier transforms of bounded bilinear forms and completely bounded multilinear operators
Noncommutative Fourier transforms of bounded bilinear forms and completely bounded multilinear operators
复制标题
有界双线性形式和完全有界多线性算子的非交换傅里叶变换
DOI：
10.1016/0022-1236(88)90034-1
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发表时间：
1988
影响因子：
1.7
通讯作者：
K. Ylinen
中科院分区：
文献类型：
--
作者：
K. Ylinen
摘要 设 G 1,…,G n 为局部紧群，H 为希尔伯特空间。任何有界 n 线性算子 Φ: C*(G 1)×···× C*(G n)→ L (H) 已知具有唯一的单独 σ-弱连续扩张 \̃ gF: W*(G 1)×···× W*(G n)→ L (H)。 Φ 的傅立叶变换是函数\̂ gF: G 1×···× G n→ L (H)，定义为\̂ gF (s 1,…, s n)=\̃ gF (ω 1 (S 1),…, ω n (s n))，其中 ω i: G i→ W*(G i)⊂ L (H ωi) 是通用表示。在 n= 2、L (H)= C 的情况下，根据 Grothendieck-Pisier-Haagerup 不等式，根据弱协调和半齐次随机场以及连续酉 Jordan 表示，给出了此类傅里叶变换的表征。在任意 n 的情况下，注意力仅限于完全有界的 n 线性 L (H) 值运算符。它们的傅里叶变换被表征，并且它们的卷积运算被定义。在 L (H)= C 的情况下，C*(G 1)×···× C*(G n) 上的完全有界 n 线性形式形成可交换的 Banach 代数，其最大理想空间包含 Δ (B (G 1))×···× Δ (B (G n)) 在具有连续左逆的单独连续注入下的图像，其中 Δ (B (G i)) 是 Fourier-Stieltjes 的最大理想空间G i 的代数。
Abstract Let G 1,…, G n be locally compact groups and H a Hilbert space. Any bounded n-linear operator Φ: C∗(G 1)×···× C∗(G n)→ L (H) is known to have a unique separately σ-weakly continuous extension\̃ gF: W∗(G 1)×···× W∗(G n)→ L (H). The Fourier transform of Φ is the function\̂ gF: G 1×···× G n→ L (H) defined by\̂ gF (s 1,…, s n)=\̃ gF (ω 1 (S 1),…, ω n (s n)), where ω i: G i→ W∗(G i)⊂ L (H ωi) is the universal representation. In the case n= 2, L (H)= C, characterizations based on the Grothendieck-Pisier-Haagerup inequality are given for such Fourier transforms in terms of weakly harmonizable and hemihomogeneous random fields and continuous unitary Jordan representations. In the case of arbitrary n, attention is confined to completely bounded n-linear L (H)-valued operators. Their Fourier transforms are characterized and a convolution operation for them is defined. In the case of L (H)= C, the completely bounded n-linear forms on C∗(G 1)×···× C∗(G n) form a commutative Banach algebra whose maximal ideal space contains the image of Δ (B (G 1))×···× Δ (B (G n)) under a separately continuous injection with a continuous left inverse, where Δ (B (G i)) is the maximal ideal space of the Fourier-Stieltjes algebra of G i.