COMPLETE BOUNDEDNESS OF HEAT SEMIGROUPS ON THE VON NEUMANN ALGEBRA OF HYPERBOLIC GROUPS.

We prove that λg → e-t|g|r λg defines a multiplier on the von Neuman algebra of hyperbolic groups with a complete bound = r, for any 0 < t < ∞, 1 < r < ∞. In the proof we observe that a construction of Ozawa allows us to characterize the radial multipliers that are bounded on every hyperbolic graph, partially generalizing results of Haagerup-Steenstrup-Szwarc and Wysocza'nski. Our argument is also based on the work of Peller. [ABSTRACT FROM AUTHOR]