Simplicity of reduced group Banach algebras

TL;DR

This paper proves the simplicity of reduced group L^p operator algebras and related Banach algebras for discrete groups with simple reduced group C*-algebras, using interpolation and existing results.

Contribution

It establishes the simplicity of reduced group L^p operator algebras and Banach algebras generated by regular representations for certain classes of groups, extending known results.

Findings

Reduced group L^p operator algebras are simple for groups with simple reduced C*-algebras.

Banach algebras generated by regular representations on reflexive Orlicz and Lorentz spaces are also simple.

Results extend to the unique trace property for these 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.