arXiv 1208.37912012-08math.FAmath.OA

某些加权群代数是算子代数

Some weighted group algebras are operator algebras

Hun Hee Lee, Ebrahim Samei, Nico Spronk

备注：: Errors concerning Grothendieck's inequality are fixed. The related constants appearing in the results are corrected accordingly

摘要：

令$G$是具有多项式增长的随机生成群，并且令$\om$是权重，即$G$上具有正值的子乘法函数。研究了加权群代数$\ell^1(G,\om)$与算子代数同构的条件。证明了如果$\om$是次数足够大的多项式权或$0<α<1$阶的指数权，则$\ell^1(G,\om)$同构于算子代数.我们将证明$G$的生长顺序在这个问题中起着重要的作用。此外，$\ell^1(G,\om)$的代数中心同构于一个$Q$ -代数，因此满足一个多变量von Neumann不等式.当$G$是$d$ -维整数$\Z^d$和三维离散Heisenberg群$\mathbb{H}_3(\Z)$时，我们也给出了我们的结果的更详细的研究.有两个生成元的自由群的情况将被认为是指数增长群的反例。

英文摘要：

Let $G$ be a finitely generated group with polynomial growth, and let $\om$ be a weight, i.e. a sub-multiplicative function on $G$ with positive values. We study when the weighted group algebra $\ell^1(G,\om)$ is isomorphic to an operator algebra. We show that $\ell^1(G,\om)$ is isomorphic to an operator algebra if $\om$ is a polynomial weight with large enough degree or an exponential weight of order $0<α<1$. We will demonstrate the order of growth of $G$ plays an important role in this question. Moreover, the algebraic centre of $\ell^1(G,\om)$ is isomorphic to a $Q$-algebra and hence satisfies a multi-variable von Neumann inequality. We also present a more detailed study of our results when $G$ is the $d$-dimensional integers $\Z^d$ and 3-dimensional discrete Heisenberg group $\mathbb{H}_3(\Z)$. The case of the free group with two generators will be considered as a counter example of groups with exponential growth.

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摘要：

英文摘要：

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