Induced $C^ *$-algebras and Landstad duality for twisted coactions

Abstract:

Suppose $N$ is a closed normal subgroup of a locally compact group $G$. A coaction $:A \to M(A \otimes {C^ * }(N))$ of $N$ on a ${C^ * }$-algebra $A$ can be inflated to a coaction $\delta$ of $G$ on $A$, and the crossed product $A{ \times _\delta }G$ is then isomorphic to the induced ${C^ * }$-algebra $\text {Ind}_N^G A{\times _\epsilon }N$. We prove this and a natural generalization in which $A{ \times _\epsilon }N$ is replaced by a twisted crossed product $A{ \times _{G/N}}G$; in case $G$ is abelian, we recover a theorem of Olesen and Pedersen. We then use this to extend the Landstad duality of the first author to twisted crossed products, and give several applications. In particular, we prove that if \[ 1 \to N \to G \to G/N \to 1\] is topologically trivial, but not necessarily split as a group extension, then every twisted crossed product $A{ \times _{G/N}}G$ is isomorphic to a crossed product of the form $A \times N$.