Crossed products of $L^p$ operator algebras and the K-theory of Cuntz algebras on $L^p$ spaces

Abstract

For $p \in [1, \infty),$ we define and study full and reduced crossed products of algebras of operators on $\sigma$-finite $L^p$ spaces by isometric actions of second countable locally compact groups. We give universal properties for both crossed products. When the group is abelian, we prove the existence of a dual action on the full and reduced $L^p$ operator crossed products. When the group is discrete, we construct a conditional expectation to the original algebra which is faithful in a suitable sense. For a free action of a discrete group on a compact metric space $X,$ we identify all traces on the reduced $L^p$ operator crossed product, and if the action is also minimal we show that the reduced $L^p$ operator crossed product is simple. We prove that the full and reduced $L^p$ operator crossed products of an amenable $L^p$ operator algebra by a discrete amenable group are again amenable. We prove a Pimsner-Voiculescu exact sequence for the K-theory of reduced $L^p$ operator crossed products by ${\mathbb{Z}}.$ We show that the $L^p$ analogs ${\mathcal{O}}_d^p$ of the Cuntz algebras ${\mathcal{O}}_d$ are stably isomorphic to reduced $L^p$ operator crossed products of stabilized $L^p$ UHF algebra by ${\mathbb{Z}},$ and show that $K_0 ({\mathcal{O}}_d^p) \cong {\mathbb{Z}} / (d - 1) {\mathbb{Z}}$ and $K_1 ({\mathcal{O}}_d^p) = 0.$