arXiv 1410.19682014-10math.OA

局部紧量子群对角逼近的希尔伯特空间方法

A Hilbert space approach to approximate diagonals for locally compact quantum groups

Benjamin Willson

DOI：: 10.15352/bjma/09-3-18

摘要：

对于局部紧量子群$\mathbb{G}$，量子群代数$L^1(\mathbb{G})$是算子顺从的当且仅当它有一个算子有界的近似对角.已知如果$L^1(\mathbb{G})$是算子双平坦的并且有一个有界的近似恒等式，那么它是算子顺从的。在本文中，我们考虑网$L^2(\mathbb{G})$足以显示这两个条件，并将它们结合起来，使一个近似对角线的形式$ω_{{W'}^*ξ\otimesη}$其中$W$是乘法酉和$ξ\otimesη$是简单的张量在$L^2(\mathbb{G})\otimes L^2(\mathbb{G})$。事实上，如果$\mathbb{G}$和$\hat{\mathbb{G}}$都有一个有界近似恒等式，并且$L^2(\mathbb{G})$中的任何一个相应的网满足推广拟中心性的条件，那么这种构造生成一个算子有界近似对角。在经典群情形下，这为构造近似对角线提供了一种新的方法，强调了群代数$L^1(G)$和傅立叶代数$A(G)$的算子顺从性之间的关系.

英文摘要：

For a locally compact quantum group $\mathbb{G}$, the quantum group algebra $L^1(\mathbb{G})$ is operator amenable if and only if it has an operator bounded approximate diagonal. It is known that if $L^1(\mathbb{G})$ is operator biflat and has a bounded approximate identity then it is operator amenable. In this paper, we consider nets in $L^2(\mathbb{G})$ which suffice to show these two conditions and combine them to make an approximate diagonal of the form $ω_{{W'}^*ξ\otimesη}$ where $W$ is the multiplicative unitary and $ξ\otimesη$ are simple tensors in $L^2(\mathbb{G})\otimes L^2(\mathbb{G})$. Indeed, if $\mathbb{G}$ and $\hat{\mathbb{G}}$ both have a bounded approximate identity and either of the corresponding nets in $L^2(\mathbb{G})$ satisfies a condition generalizing quasicentrality then this construction generates an operator bounded approximate diagonal. In the classical group case, this provides a new method for constructing approximate diagonals emphasizing the relation between the operator amenability of the group algebra $L^1(G)$ and the Fourier algebra $A(G)$.

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