Objective
Von Neumann's concept of quantization goes back to the foundations of quantum mechanics
and provides a noncommutative model of integration. Over the years, von Neumann algebras
have shown a profound structure and set the right framework for quantizing portions of algebra,
analysis, geometry and probability. A fundamental part of my research is devoted to develop a
very much expected Calderón-Zygmund theory for von Neumann algebras. The lack of natural
metrics partly justifies this long standing gap in the theory. Key new ingredients come from
recent results on noncommutative martingale inequalities, operator space theory and quantum
probability. This is an ambitious research project and applications include new estimates for
noncommutative Riesz transforms, Fourier and Schur multipliers on arbitrary discrete groups
or noncommutative ergodic theorems. Other related objectives of this project include Rubio
de Francia's conjecture on the almost everywhere convergence of Fourier series for matrix
valued functions or a formulation of Fefferman-Stein's maximal inequality for noncommutative
martingales. Reciprocally, I will also apply new techniques from quantum probability in
noncommutative Lp embedding theory and the local theory of operator spaces. I have already
obtained major results in this field, which might be useful towards a noncommutative form of
weighted harmonic analysis and new challenging results on quantum information theory.
Fields of science
Call for proposal
ERC-2010-StG_20091028
See other projects for this call
Funding Scheme
ERC-SG - ERC Starting GrantHost institution
28006 Madrid
Spain