Representations of crossed products by coactions and principal bundles

Abstract

The main purpose of this paper is to establish a covariant representation theory for coactions of locally compact groups on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> ∗ </mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{C^{\ast }}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebras (including a notion of exterior equivalence), to show how these results extend the usual notions for actions of groups on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> ∗ </mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{C^{\ast }}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebras, and to apply these ideas to classes of coactions termed <italic>pointwise unitary</italic> and <italic>locally unitary</italic> to obtain a complete realization of the isomorphism theory of locally trivial principal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -bundles in this context. We are also able to obtain all (Cartan) principal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -bundles in this context, but their isomorphism theory remains elusive.