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

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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.