Generalisations of the Haagerup approximation property to arbitrary von Neumann algebras
Generalisations of the Haagerup approximation property to arbitrary von Neumann algebras
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DOI：
10.1016/j.crma.2014.04.003
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发表时间：
2014-04
影响因子：
0.8
通讯作者：
Martijn Caspers;Rui Okayasu;Adam G. Skalski;Reiji Tomatsu
中科院分区：
文献类型：
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作者：
Martijn Caspers;Rui Okayasu;Adam G. Skalski;Reiji Tomatsu
局部紧群的Haagerup性质起源于著名的文献[9]，其中U。Haagerup证明了自由群上的长度函数是条件负定的。这一事实，即在一个群G上存在一个真连续的条件负定函数，可以被看作是G的顺从性的自然松弛，后来被证明是几何群论、动力系统和算子代数中一个自然而有影响力的概念（见[3]）。1983年，M. Choda在[4]中证明，如果G是离散的，则G具有Haagerup性质当且仅当G的群von Neumann代数具有其自然迹，具有一定的von Neumann代数逼近性质，这就是所谓的具有忠实正规迹态的von Neumann代数的Haagerup逼近性质。后一个性质也是由对von Neumann代数上的上循环作用的研究所激发的[5]。后来，仍然在有限冯诺依曼代数的框架下，P. Jolissaint在[10]中详细研究了它，他特别证明了它实际上不依赖于忠实正常迹态的选择。公平地说，将Haagerup性质的概念扩展到无限冯诺依曼代数（或者在有限冯诺依曼代数上构建L2-近似时甚至只考虑非迹态）的兴趣在几年内相当有限，因为一般来说，没有希望通过其冯诺依曼代数来证明非离散群的Haagerup性质。同样的，G的顺从性和VN（G）的内射性之间的关系对于非离散群也不成立。在过去的几年里，随着对离散和局部紧量子群的Haagerup性质的研究的出现，这种情况发生了巨大的变化。它是由M. [1]，然后继续在几个方向（见[6]和其中的参考文献）。关键在于离散量子群的von Neumann代数的自然（Haar）态不需要是迹的，因此，要通过离散量子群的von Neumann代数证明其Haagerup性质，首先需要理解在没有迹态的情况下冯·诺依曼代数Haagerup近似的性质。方向：MC和AS在[2]中研究了原始定义的直接变体，基于近似完全正映射的存在性，这些映射对于给定的忠实正规权表现良好，并且其L2-GNS实现是紧凑的，而RO和RT在[12]中工作在标准形式的框架中，部分受到A.鱼雷[16]。在这些预印本的原始版本流传之后，本文作者观察到，事实上这两种方法导致了等同的概念（尽管这只是在各自的理论充分发展之后才变得清晰）。本文总结了关于任意von Neumann代数的Haagerup性质的主要结果，并参考文献[2]和[12]对这些结果进行了证明。
The Haagerup property for a locally compact group has its origins in the celebrated paper [9], where U. Haagerup proved that the length function on the free group is conditionally negative definite. This fact, ie the existence on a group G of a proper continuous conditionally negative definite function can be viewed as a natural relaxation of the amenability of G and has later proved to be a natural and influential notion in geometric group theory, dynamical systems, and operator algebras (see [3]). In particular, already in 1983, M. Choda showed in [4] that if G is discrete, then it has the Haagerup property if and only if the group von Neumann algebra of G equipped with its natural trace has a certain von Neumann algebraic approximation property, which came to be known as the Haagerup approximation property for a von Neumann algebra with a faithful normal tracial state. The latter property was also motivated by the study of cocycle actions on von Neumann algebras [5]. Later, still in the framework of finite von Neumann algebras, it was studied in detail by P. Jolissaint in [10], where he proved in particular that in fact it does not depend on the choice of the faithful normal tracial state. It is fair to say that the interest in extending the notion of the Haagerup property to infinite von Neumann algebras (or even just considering non-tracial states when building L2-approximations on finite von Neumann algebras) was rather limited for several years due to the fact that in general there is no hope to characterise the Haagerup property for a non-discrete group via its von Neumann algebra. In the same spirit, the relation between the amenability of G and the injectivity of VN (G) breaks down for non-discrete groups. This changed drastically in the last few years, with the advent of a study of the Haagerup property for discrete and locally compact quantum groups. It was initiated by the study of the dual of the quantum free orthogonal and free unitary group by M. Brannan [1] and then continued in several directions (see [6] and references therein). The key factor lies in the fact that the natural (Haar) states of the von Neumann algebras of discrete quantum groups need not be tracial, so, to characterise the Haagerup property for a discrete quantum group via its von Neumann algebra, one first needs to develop an understanding of the von Neumann algebraic Haagerup approximation property in the absence of a tracial state.The authors of this note approached this question from two directions: MC and AS in [2] studied a direct variant of the original definition, based on the existence of the approximating completely positive maps which behave well with respect to a given faithful normal weight and whose L2-GNS implementations are compact, whereas RO and RT in [12] worked in the framework of standard forms, partly motivated by the approach to injectivity due to A. Torpe [16]. After the original versions of these preprints were circulated, the present authors observed that in fact the two approaches led to equivalent notions (although this only became clear after the respective theories were fully developed). In this note, we summarise the main results regarding the Haagerup property for arbitrary von Neumann algebras, referring to [2] and [12] for the proofs of the presented statements.