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

这个建议是关于三个领域的数学是紧密交织在一起：算子空间，局部紧量子群，并服从Banach代数。 顺从Banach代数是由B引入的。E. 1972年的约翰逊：之所以使用这个术语是因为约翰逊的定理，该定理断言局部紧群G是顺从的当且仅当它的群代数L^1（G）是顺从的Banach代数。另一方面，在“对偶”方面，一个类似的定理失败了：有顺从的--甚至是紧的--群G，对于这些群，傅里叶代数A（G）是不顺从的。为了得到类似于约翰逊的结果，必须考虑A（G）的自然算子空间结构.事实上，Z。J. Ruan证明了，G是顺从的当且仅当A（G）是算子顺从的. L^1（G）和A（G）之间的非形式对偶可以在局部紧量子群的框架下形式化，在J. Kustermans和S.维斯同样，当G是局部紧量子群（简称：LCQG）时，算子空间方法在L^1（G）的研究中起着重要的作用。关于LCQG的顺从性有各种各样的概念，我们的部分研究是研究这些概念与L^1（G）的（算子）顺从性之间的关系。例如，直到最近还有一个似是而非的猜想，即L^1（G）是算子顺从的当且仅当G是顺从的且共同顺从的：这样的结果将包含约翰逊和阮的定理作为特例。唉，M.卡斯珀斯H. Lee和E. Ricard证明了L^1（G）的算子双平坦性已经迫使G是Kac型的。当然，这就引出了一个问题，即上述猜想对于Kac型的LCQG是否最不成立。无论如何，LCQG的性质和L^1（G）的算子顺从性（以及其他上同调性质）之间的关系比看起来的要微妙得多，我们计划详细地研究它。 当然，顺从Banach代数理论的应用远远超出群代数及其量子推广。关于Banach空间E上所有有界线性算子的Banach代数B（E），在过去的几年里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。2009年，S。A. Argyros和R. G.海顿解决了所谓的“标量加紧问题”，并作为副产品，他们获得了在无限维Banach空间E，其中B（E）是服从。另一方面，长期存在的猜想B（l^p）对任何p都是不成立的，得到了肯定的解决。这就引出了代数B（E）适用于哪个Banach空间E的问题。另外，如果E是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）的顺从性是否必然导致B（E）/K（E）是有限维的？我们将研究这些（和其他）问题。 最后，除了它们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。我们特别感兴趣的概念，紧性和弱紧性的运营商空间的背景。算子空间的紧性有各种各样的概念，但似乎没有比Banach空间意义下的弱紧性更强的算子空间的弱紧性概念。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，因为几乎和弱几乎周期性等概念可以通过紧性和弱紧性的算子空间概念来适应A（G）（或L^1（G）对于LCQG G）。

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.