On commutative, operator amenable subalgebras of finite von Neumann algebras
On commutative, operator amenable subalgebras of finite von Neumann algebras
复制标题
关于有限冯·诺依曼代数的交换算子顺应子代数
DOI：
10.1515/crelle.2012.030
复制
发表时间：
2010
期刊：
影响因子：
--
通讯作者：
Yemon Choi
中科院分区：
文献类型：
--
作者：
Yemon Choi
根据Dixmier和Day关于可调群的有界Hilbertian表示的旧结果，我们推测，每一个规范闭的可调子代数都与可调C*-代数自动相似。Curtis和Loy（1995）、Gifford（2006）和Marcoux（2008）的结果提供了一些证据来支持这一猜想，但它仍然对交换子代数开放。我们提出了更多的证据来支持这一猜想，通过证明一个封闭的，交换的，算子可服从的有限冯诺伊曼代数的子代数必须类似于一个自伴随子代数。所使用的技术结果包括基于Grothendieck不等式和Pietsch支配定理的近似论证，以及对Gifford定理的一个适应，该定理适用于附着于<s:1>无界算子的设置。
Abstract It has been conjectured, motivated in part by old results of Dixmier and Day on bounded Hilbertian representations of amenable groups, that every norm-closed amenable subalgebra of ℬ(ℋ) is automatically similar to an amenable C*-algebra. Results of Curtis and Loy (1995), Gifford (2006), and Marcoux (2008) give some evidence to support this conjecture, but it remains open even for commutative subalgebras. We present more evidence to support this conjecture, by showing that a closed, commutative, operator amenable subalgebra of a finite von Neumann algebra ℳ must be similar to a selfadjoint-subalgebra. Technical results used include an approximation argument based on Grothendieck's inequality and the Pietsch Domination Theorem, together with an adaptation of a theorem of Gifford to the setting of unbounded operators affiliated to ℳ.