Noncommutative analysis on groups and quantum groups – ANCG

The consortium consists of researchers around noncommutative Lpspaces,
operator spaces/algebras and quantum groups. Multipliers play a central role in their research and seem to be a perfect matching point to create more and stronger links.
The project will give an opportunity to bring closer operator algebraists and functional analysts with expertise from different fields. There exist few places working in these different directions at the same
time in the world. The project will allow partners to visit each other and to collaborate with foreigners. Postdoctoral fellowships will be a motor of deepening links between researchers. The partners will also complete
the project by PhD fellowships by other funding.

Noncommutative integration is in the heart of many quantized theories in mathematics and physics. Basically, in these theories, the idea is to drop the commutativity assumption for function algebras, so functions become operators. Noncommutative integration really took a new start after the development of operator spaces in 1990s which gives a perfect setting and new tools for studying problems of analytic nature in quantized theories. From its roots, the quantized world has close links with operator algebras. Indeed, noncommutative analysis finds one of its origins in the investigation of approximation properties of operator algebras in the 1980s. It also appears to be the right point of view for dealing with many subjects from other quantized theories such as noncommutative geometry and quantum information. For instance, Connes’ quantized differential calculus and its recent extension to a truly noncommutative setting by Sukochev’s team are based on pseudo-differential operators on noncommutative Sobolev spaces. On the other hand, the exploration of quantum information by noncommutative analysis is a very fertile area as shown by the recent remarkable developments around quantum entropy by means of noncommutative functional inequalities. Fourier and Schur multipliers are at the intersection of these areas and play a crucial role in recent researches motivated by concepts and problems from operator algebras and geometric group theory. The first interactions between operator spaces, operator algebras and quantum probability are already impressive. We confidently expect that their full exploration will open up new perspectives and significantly impact the future development of these areas.

Fourier multipliers are by far the most important operators in classical analysis. Indeed, one can say that classical harmonic analysis has been developed around Fourier multipliers. The main task there is to find criteria for the boundedness of multipliers on various function spaces, notably on Lp-spaces. Although noncommutative analysis is a very new topic, the theory of noncommutative Lp-spaces has a long history going back to pioneering works of Schatten/von Neumann, Dixmier and Segal in the tracial case. Based on Tomita-Takesaki’s modular theory, the generalization to type III algebras was achieved later by Connes/Hilsum, Haagerup and Kosaki. The final step was done by Pisier who introduced their operator space structure. The preeminent place occupied by Fourier multipliers in classical analysis leads one to expect a similar strong role to be played by their counterparts, Fourier and Schur multipliers in noncommutative analysis. However, to date there are very few results in the literature on the boundedness of Fourier and Schur multipliers on noncommutative Lp-spaces for finite p. There are multiple reasons for this: noncommutative integration is hard to manage; many classical techniques and tools involving maximal functions and stopping times are no longer available in the noncommutative setting. Nevertheless, research on Fourier and Schur multipliers on noncommutative Lp-spaces has already produced some success, a noncommutative Calderón-Zygmund theory is emerging. However, it still has to be pushed one step further to reach the richness of classical analysis.

The primary objective of this project is to develop an exhaustive study of noncommutative multipliers and to seek their applications in noncommutative harmonic analysis in a wide sense. An ambitious part is to establish Hörmander-Mikhlin type theorems for radial multipliers on certain groups, in particular, on free groups. This would lead to the full realization of a noncommutative Calderón-Zygmund theory. Another fundamental aspect concerns applications to approximation properties, a challenging problem is to exhibit a concrete Lp-space without the Grothendieck or completely bounded approximation property for all p>2; this is a longstanding open problem in the domain.