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 $K_1({\mathcal{O}}_d^p)=0.$

AMSLaTeX; 54 pages

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