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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=588841
• 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.

这一建议涉及数学中紧密交织的三个领域：算子空间、局部紧量子群和服从Banach代数。 自适应Banach代数是由B.E.Johnson在1972年提出的：这个术语的由来是Johnson定理，它断言一个局部紧群G是自适应的当且仅当它的群代数L^1(G)是自适应的Banach代数。另一方面，在“对偶”方面，类似的定理也不成立：存在傅立叶代数A(G)不服从的、甚至是紧的-群G。为了得到与Johnson的结果类似的结果，必须考虑A(G)的自然算子空间结构。事实上，正如Z.J.Ruan所证明的，G是顺从的当且仅当A(G)是顺从的算子。 L^1(G)和A(G)之间的非正式对偶可以在J.Kustermann和S.Vaes意义下的局部紧量子群的框架下形式化。再次，当G是局部紧量子群时，算符空间方法在L^1(G)的研究中起着重要的作用。关于可修饰性的概念有很多种，本研究的一部分是调查这些概念与L^1(G)的(操作员)可修饰性之间的关系。例如，直到最近，一个似是而非的猜想仍然是L^1(G)是算子服从的当且仅当G既服从又共服从：这样的结果将包含Johnson和Ruan的定理作为特例。遗憾的是，M.Caspers，H.H.Lee和E.Ricard最近的一篇论文表明，L^1(G)的算子双平面性已经迫使G是Kac型的。当然，这引出了一个问题，即上述猜想是否对Kac类型的LCQG是最不正确的。无论如何，LcQG G的性质与L^1(G)的算子顺从性(和其他上同调性质)之间的关系比看起来要微妙得多，我们计划对其进行详细的研究。 当然，服从Banach代数的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)只对有限维E起作用。然后，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上得到了B(E)起作用的结果。另一方面，长期存在的关于B(L^p)对任何p都不可服从的猜想得到了肯定的解决。这就引出了代数B(E)对哪个Banach空间E是服从的问题。此外，如果E是由Argyros和Haydon构造的空间，则B(E)服从B(E)/K(E)(其中K(E)是E上的紧算子)是一维的。B(E)的顺从性是否必然意味着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，算子空间本身也是有趣的数学对象。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。关于算子空间的紧性有各种不同的概念，但对于超越Banach空间意义下的弱紧性的算子空间，似乎没有弱紧性的概念。最后，这一方向的研究可能会再次对抽象调和分析产生影响，因为几乎和弱概周期等概念可以使用紧性和弱紧性的算子空间概念改写成A(G)(或者对于LCQG，L^1(G))。