Block maps and Fourier analysis

TL;DR

This paper introduces block maps for subfactors, analyzes their dynamics, and connects these to Fourier analysis, additive combinatorics, and classical inequalities, revealing new structural insights and asymptotic behaviors.

Contribution

It develops the theory of block maps for subfactors, proves inverse sum set theorems, and characterizes extremal operators, extending Fourier analysis in this context.

Findings

Limit points of the dynamic system are positive multiples of biprojections and zero.

For Z2 case, block map behavior matches that of the 2D Ising model.

Established inverse sum set theorem for subfactors.

Abstract

We introduce block maps for subfactors and study their dynamic systems. We prove that the limit points of the dynamic system are positive multiples of biprojections and zero. For the Z2 case, the asymptotic phenomenon of the block map coincides with that of that 2D Ising model. The study of block maps requires a further development of the recent work of the authors on the Fourier analysis of subfactors. We generalize the notion of sum set estimates in additive combinatorics for subfactors and prove the exact inverse sum set theorem. Using this new method, we characterize the extremal pairs of Young's inequality for subfactors, as well as the extremal operators of the Hausdorff-Young inequality.