UAlberta PIMS Mathematics and Statistics Colloquium: Matthias Neufang

 Locally compact quantum groups, as introduced by Kustermans–Vaes in 2000, provide a powerful abstract framework in which the convolution algebra L1(G) over a locally compact group G is the (Pontryagin-type) dual object of the Fourier algebra A(G); moreover, this setting encompasses important non group-like algebras arising from mathematical physics. In joint work, we have discovered that quantum group duality leads to intriguing new Banach algebra structures on the space of trace class operators T (L2(G)) associated with a locally compact quantum group G. After outlining some basic quantum group theory, I shall give a survey of our corresponding results and various applications. Indeed, T (L2(G)) carries two dual products which, in the group case, are operator analogues of the convolution product in L1(G) and the pointwise product in A(G), respectively. We study in particular the corresponding
module structures on the space B(L2(G)) of all bounded linear operators. As an application, we
characterize quantum group amenability in terms of injectivity in the category of T (L2(G))-modules. I shall also present our very recent result that the space T (L2(G))0 of trace class operators of trace 0 carries naturally a mixed Lie-type product, stemming from the two dual products, that is associative: this gives rise to a further novel Banach algebra, capturing quantum group duality. All three products are new even in the case of matrices. We show, for instance, that the resulting algebra T (L2(G))0 has a bounded approximate identity if and onlyif G and its dual G^ are both co-amenable.

The talk is mainly based on joint work with J. Crann, Z. Hu, M. Kalantar, and Z.-J. Ruan.