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For $p \in [1, \infty),$ we define and study full and reduced crossedproducts of algebras of operators on $\sigma$-finite $L^p$ spaces by isometricactions of second countable locally compact groups. We give universalproperties for both crossed products. When the group is abelian, we prove theexistence of a dual action on the full and reduced $L^p$ operator crossedproducts. When the group is discrete, we construct a conditional expectation tothe original algebra which is faithful in a suitable sense. For a free actionof a discrete group on a compact metric space $X,$ we identify all traces onthe reduced $L^p$ operator crossed product, and if the action is also minimalwe show that the reduced $L^p$ operator crossed product is simple. We provethat 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 aPimsner-Voiculescu exact sequence for the K-theory of reduced $L^p$ operatorcrossed products by ${\mathbb{Z}}.$ We show that the $L^p$ analogs${\mathcal{O}}_d^p$ of the Cuntz algebras ${\mathcal{O}}_d$ are stablyisomorphic to reduced $L^p$ operator crossed products of stabilized $L^p$ UHFalgebra by ${\mathbb{Z}},$ and show that $K_0 ({\mathcal{O}}_d^p) \cong{\mathbb{Z}} / (d - 1) {\mathbb{Z}}$ and $K_1 ({\mathcal{O}}_d^p) = 0.$

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