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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=612062
• 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.

该提案大约是数学紧密交织的三个领域：操作员空间，局部紧凑的量子群和可及的Banach代数。 B. E. Johnson于1972年引入了Banach代数：术语的原因是约翰逊定理，断言局部紧凑的G组在且仅当其组代数l^1（g）是一个可正常的Banach代数时。另一方面，在“双重”方面，类似的定理失败了：傅立叶代数A（g）不适合使用，甚至是紧凑型组的G组。为了对约翰逊的结果进行模拟，必须考虑A（g）的自然操作员空间结构。的确，正如Z. J. Ruan所证明的那样，当A（g）是操作员时，G是可以正常的。 从J. Kustermans和S. vaes的意义上讲，L^1（g）和A（g）之间的非正式二元性可以在局部紧凑型量子组的框架中形式化。同样，当G是局部紧凑的量子组（总的来说：LCQG）时，操作员空间方法在L^1（g）的研究中起重要作用。 LCQG有各种各样的概念，拟议的研究的一部分是研究这些概念如何与L^1（g）的（操作员）合理性有关。例如，直到最近，当且仅当G既适合且可共同）时，l^1（g）是可行的。 las，M。Caspers，H。H。Lee和E. Ricard最近的一篇论文表明，L^1（g）强迫G的操作员Biflatness g属于KAC类型。当然，这要解决上述猜想是否对KAC类型的LCQG是至少正确的问题。无论如何，LCQG G的性质与L^1（g）的操作员的合理性（以及其他同胞特性）之间的关系比看起来更加微妙，我们计划详细研究它。 当然，Amenable Banach代数理论的应用超出了群体代数及其量子概括。 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 B（e）的无限尺寸BANACH空间E是可正常的。另一方面，B（L^p）的长期猜想是没有肯定地解决的。这导致了一个问题，即代数B（e）的Banach空间可正常。同样，如果E是Argyros和Haydon构建的空间，则B（e）可以正常b（e）/k（e）（其中k（e）是e）上紧凑的操作员是一维的。 b（e）的舒适性是否必然需要b（e）/k（e）是有限维度的？我们计划调查这些（和其他）问题。 最后，除了将其用于抽象谐波分析的适用性之外，操作员空间本身就是数学对象。我们对对操作员空间环境的紧凑性和紧凑性弱的概念特别感兴趣。操作员空间有各种紧凑的概念，但是在Banach空间感觉中，似乎没有针对操作员空间的紧凑型弱的概念。最后，朝这个方向的调查可能会再次对抽象的谐波分析产生影响，因为使用操作员空间概念的紧凑型和弱的紧凑性，可以将几乎和弱的周期性等概念适应（g）（g）（g）（l^1（g））。