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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Authors: John C. Quigg and Iain Raeburn
Journal: Trans. Amer. Math. Soc. 347 (1995), 2885-2915
MSC: Primary 46L55; Secondary 46L40
MathSciNet review: 1297536
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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 $A \times N$.

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Article copyright: © Copyright 1995 American Mathematical Society