Simplicity of UHF and Cuntz algebras on $L^p$~spaces (1309.0115v2)
Abstract: We prove that, for $p \in [1, \infty),$ and integers $d$ at least 2, the $Lp$ analog ${\mathcal{O}}_dp$ of the Cuntz algebra ${\mathcal{O}}_d$ is a purely infinite simple amenable Banach algebra. The proof requires what we call the spatial $Lp$ UHF algebras, which are analogs of UHF algebras acting on $Lp$ spaces. As for the usual UHF C*-algebras, they have associated supernatural numbers. For fixed $p \in [1, \infty),$ we prove that any spatial $Lp$ UHF algebra is simple and amenable, and that two such algebras are isomorphic if and only if they have the same supernatural number (equivalently, the same scaled ordered $K_0$-group). For distinct $p_1, p_2 \in [1, \infty),$ we prove that no spatial $L{p_1}$ UHF algebra is isomorphic to any spatial $L{p_2}$ UHF algebra.