An L^p-Imprimitivity Theorem for the Group of Integers

Abstract

In this paper, the authors obtain an L^p-version of Green’s Imprimitivity Theorem for the group ℤ of integers. More precisely, for X = ℤ/nℤ and the action α of ℤ on X by translation, it is proved that the full L^p-operator crossed product F^p(ℤ, X, α) is isometrically isomorphic to the spatial L^p-operator tensor product of M_n^p and the reduced group L^p-operator algebra F_λ^p(ℤ). This solves the L^p-imprimitivity problem raised by Phillips for the group of integers. They also prove that F^p(ℤ, X, α) is isometrically isomorphic to C(S¹, M_n^p) precisely when p = 2, where S¹ denotes the unit circle in the complex plane ℂ. Moreover, they determine the K-theory groups of F^p(ℤ, X, α).

Keywords

L^p-operator crossed products / L^p-operator algebras / Gelfand transform / K-theory / 47L55 / 47L65 / 19K99 / 46J05

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