Operator spaces, locally compact quantum groups, and amenable Banach algebras

算子空间、局部紧量子群和适用的巴纳赫代数

• 批准号: RGPIN-2014-06155
• 负责人: Runde, Volker
• 金额: $ 1.31万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=567626
• 关键词: Operator; spaces; locally; compact; quantum

This proposal is about three areas in mathematics that are closely interwoven: operator spaces, locally compact quantum groups, and amenable Banach algebras. Amenable Banach algebras were introduced by B. E. Johnson in 1972: the reason for the terminology is Johnson's theorem that asserts that a locally compact group G is amenable if and only if its group algebra L^1(G) is an amenable Banach algebra. On the other hand, on the "dual" side a similar theorem fails: there are amenable - even compact - groups G for which the Fourier algebra A(G) is NOT amenable. In order to get an analog of Johnson's result one has to take the natural operator space structure of A(G) into account. Indeed, as Z. J. Ruan proved, G is amenable if and only if A(G) is operator amenable. The informal duality between L^1(G) and A(G) can be formalized in the framework of locally compact quantum groups in the sense of J. Kustermans and S. Vaes. Again, operator space methods play an important role in the study of L^1(G) when G is a locally compact quantum group (in short: LCQG). There are various notions of amenability for LCQGs, and part of the proposed research is to investigate how these notions relate to the (operator) amenability of L^1(G). For instance, until recently it was a plausible conjecture that L^1(G) is operator amenable if and only if G is both amenable and co-amenable: such a result would contain the theorems by Johnson and Ruan as special cases. Alas, a very recent paper by M. Caspers, H. H. Lee, and E. Ricard shows that already the operator biflatness of L^1(G) forces G to be of Kac type. This, of course, begets the question if the aforementioned conjecture is a least true for LCQGs of Kac type. Anyhow, the relationship between the properties of an LCQG G and the operator amenability (and other cohomological properties) of L^1(G) is more subtle than it seems, and we plan to investigate it in detail. Of course, the theory of amenable Banach algebras has applications way beyond group algebras and their quantum generalizations. Considerable progress has been achieved over the past few years with regards to the Banach algebra B(E) of all bounded linear operators on a Banach space E. It had long been believed that B(E) is amenable only for finite-dimensional E. Then, in 2009, S. A. Argyros and R. G. Haydon solved the so-called "scalar-plus-compact problem", and as a by-product, they obtained in infinite-dimensional Banach space E for which B(E) is amenable. On the other hand, the long standing conjecture that B(l^p) is not-amenable for any p was settled affirmatively. This leads to the question for which Banach spaces E the algebra B(E) is amenable. Also, if E is the space constructed by Argyros and Haydon, then B(E) is amenable that B(E)/K(E) (where K(E) are the compact operators on E) is one-dimensional. Does the amenability of B(E) necessarily entail that B(E)/K(E) is finite-dimensional? We plan to investigate these (and other) questions. Finally, beyond their applicability to abstract harmonic analysis, operator spaces are interesting as mathematical objects in their own right. We are particularly interested in notions of compactness and weak compactness to the operator space context. There are various notions of compactness for operator space, but there seems to be no notion of weak compactness for operator space that goes beyond weak compactness in the Banach space sense. In the end, investigations in this direction will likely have again repercussions to abstract harmonic analysis as notions like almost and weak almost periodicity can be adapted to A(G) (or L^1(G) for a LCQG G) using operator space notions for compactness and weak compactness.

这一建议是关于数学的三个领域紧密交织：算子空间，局部紧量子群，和可调节的巴纳赫代数。可调巴纳赫代数是由b.e. Johnson在1972年引入的：使用这个术语的原因是Johnson定理，该定理断言一个局部紧群G是可调的当且仅当它的群代数L^1(G)是可调巴纳赫代数。另一方面，在“对偶”方面，一个类似的定理失效了：存在傅里叶代数a (G)不适用的可适应-甚至紧化-群G。为了得到类似Johnson的结果，我们必须考虑到A(G)的自然算子空间结构。的确，正如阮正杰所证明的，当且仅当A(G)是算子可令的，G是可令的。L^1(G)和A(G)之间的非正式对偶性可以在J. Kustermans和S. Vaes意义上的局部紧量子群框架中形式化。当G是一个局部紧量子群（即：立法会G）时，算子空间方法在L^1(G)的研究中再次发挥了重要作用。立法会G题的适任性有不同的概念，建议研究的一部分是探讨这些概念如何与L^1(G)的（算子）适任性有关。例如，直到最近，当且仅当G既可调又共可调时，L^1(G)是算子可调的，这是一个似是而非的猜想：这样的结果将包含Johnson和阮的定理作为特例。唉，M. Caspers， H. H. Lee和E. Ricard最近的一篇论文表明，L^1(G)的算子双平坦性已经迫使G为Kac型。当然，这就产生了一个问题，即上述猜想是否至少适用于Kac类型的立法会g题。无论如何，一个立法会G G的性质与L^1(G)的算子适性（和其他上同性质）之间的关系比它看起来更微妙，我们计划详细研究它。当然，可服从巴拿赫代数理论的应用远远超出了群代数及其量子推广。在过去的几年里，关于Banach空间E上所有有界线性算子的Banach代数B(E)已经取得了相当大的进展。长期以来，人们一直认为B(E)只适用于有限维E。然后，在2009年，S. a . Argyros和R. G. Haydon解决了所谓的“标量加紧问题”，作为副产品，他们在无限维Banach空间E中得到了B(E)适用的结果。另一方面，长期存在的关于B（l^p）对任何p都不适用的猜想被肯定地解决了。这就引出了巴拿赫空间E和代数B(E)是否适用的问题。同样，如果E是由Argyros和Haydon构造的空间，则B(E)可以满足B(E)/K(E)（其中K(E)是E上的紧算子）是一维的。B(E)的适应性是否必然意味着B(E)/K(E)是有限维的？我们计划调查这些（和其他）问题。最后，除了适用于抽象调和分析之外，算子空间本身作为数学对象也是很有趣的。我们对算子空间上下文的紧性和弱紧性的概念特别感兴趣。对于算子空间有各种各样的紧性概念，但是对于算子空间来说，似乎没有超出巴拿赫空间意义上的弱紧性的弱紧性概念。最后，在这个方向上的研究可能会再次对抽象调和分析产生影响，因为像几乎和弱几乎周期这样的概念可以适用于A(G)（或L^1(G)对于一个立法会G G），使用算子空间的紧性和弱紧性概念。