Simplicity of reduced group Banach algebras

Abstract

Let G be a discrete group. Suppose that the reduced group C*-algebra of G is simple. We use results of Kalantar-Kennedy and Haagerup, and Banach space interpolation, to prove that, for p in (1,infinity), the reduced group L^p operator algebra F^p_r(G) and its *-analog B^{p,*}_r(G) are simple. If G is countable, we prove that the Banach algebras generated by the left regular representations on reflexive Orlicz sequence spaces and certain Lorentz sequence spaces are also simple. We prove analogous results with simplicity replaced by the unique trace property. For use in the Orlicz sequence space case, we prove that if p is in (1,infinity), then any reflexive Orlicz sequence space is isomorphic (not necessarily isometrically) to a space gotten by interpolation between l^p and some other Orlicz sequence space.