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.

该提案涉及紧密交织的三个数学领域：算子空间、局部紧量子群和适用的巴纳赫代数。 服从的巴纳赫代数由 B. E. Johnson 于 1972 年提出：该术语的原因是约翰逊定理，该定理断言局部紧群 G 是服从的当且仅当其群代数 L^1(G) 是服从的巴纳赫代数。另一方面，在“对偶”方面，类似的定理失败了：存在适合的（甚至是紧凑的）群 G，而傅立叶代数 A(G) 不适用于这些组。为了获得约翰逊结果的模拟，必须考虑 A(G) 的自然算子空间结构。事实上，正如 Z. J. Ruan 所证明的那样，当且仅当 A(G) 是操作者服从的时，G 才是服从的。 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的框架中形式化。同样，当 G 是局部紧量子群（简称：LCQG）时，算子空间方法在 L^1(G) 的研究中发挥着重要作用。 LCQG 的顺应性有多种概念，拟议研究的一部分是调查这些概念与 L^1(G) 的（算子）顺应性如何相关。例如，直到最近，还有一个合理的猜想：L^1(G) 是算子服从的，当且仅当 G 既服从又服从：这样的结果将包含 Johnson 和 Ruan 的定理作为特例。唉，M. Caspers、H. H. Lee 和 E. Ricard 最近发表的一篇论文表明，L^1(G) 的算子双平坦性已经迫使 G 为 Kac 类型。当然，这引发了一个问题：上述猜想对于 Kac 类型的 LCQG 是否最不成立。无论如何，LCQG G 的属性与 L^1(G) 的算子顺从性（以及其他上同调属性）之间的关系比看起来更微妙，我们计划对其进行详细研究。 当然，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) 仅适用于有限维 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) 是可以接受的。另一方面，长期存在的猜想 B(l^p) 不适用于任何 p 得到了肯定的解决。这就引出了代数 B(E) 适合巴纳赫空间 E 的问题。另外，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) 的适用性是否必然意味着 B(E)/K(E) 是有限维的？我们计划调查这些（和其他）问题。 最后，除了对抽象调和分析的适用性之外，算子空间本身作为数学对象也很有趣。我们对运算符空间上下文的紧致性和弱紧致性概念特别感兴趣。算子空间的紧致性有多种概念，但似乎没有一个算子空间的弱紧致性概念超出了 Banach 空间意义上的弱紧致性。最后，这个方向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性这样的概念可以使用算子空间概念适应 A(G)（或 L^1(G) 对于 LCQG G）用于紧致性和弱紧致性。