Complete boundedness of the Heat Semigroups on the von Neumann Algebra of hyperbolic groups

TL;DR

This paper establishes that certain heat semigroups are completely bounded on the von Neumann algebra of hyperbolic groups for all real exponents, using advanced techniques in operator algebras and harmonic analysis.

Contribution

It proves the complete boundedness of heat semigroups on hyperbolic group von Neumann algebras for all real powers, extending previous results and employing novel characterizations.

Findings

Complete boundedness of the semigroup for all real r

Characterization of radial multipliers on hyperbolic graphs

Upper bounds for trace class norms of Hankel matrices

Abstract

We prove that $(\lambda_g\mapsto e^{-t|g|^r}\lambda_g)_{t>0}$ defines a completely bounded semigroup of multipliers on the von Neuman algebra of hyperbolic groups for all real number $r$. One ingredient in the proof is the observation that a construction of Ozawa allows to characterize the radial multipliers that are bounded on every hyperbolic graph, partially generalizing results of Haagerup--Steenstrup--Szwarc and Wysocza\'nski. Another ingredient is an upper estimate of trace class norms for Hankel matrices, which is based on Peller's characterization of such norms.