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

We prove that $({\lambda }_g\mapsto e^{-t\mathrm{\mid }g{\mathrm{\mid }}^r}{\lambda }_g{)}_{t>0}$ 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.