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Faithful representations of crossed products by endomorphisms


Authors: Sarah Boyd, Navin Keswani and Iain Raeburn
Journal: Proc. Amer. Math. Soc. 118 (1993), 427-436
MSC: Primary 46L55
DOI: https://doi.org/10.1090/S0002-9939-1993-1126190-X
MathSciNet review: 1126190
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Abstract: Stacey has recently characterised the crossed product $ A{ \times _\alpha }{\mathbf{N}}$ of a $ {C^{\ast}}$-algebra $ A$ by an endomorphism $ \alpha $ as a $ {C^{\ast}}$-algebra whose representations are given by covariant representations of the system $ (A,\alpha )$. Following work of O'Donovan for automorphisms, we give conditions on a covariant representation $ (\pi ,S)$ of $ (A,\alpha )$ which ensure that the corresponding representation $ \pi \times S$ of $ A{ \times _\alpha }{\mathbf{N}}$ is faithful. We then use this result to improve a theorem of Paschke on the simplicity of $ A{ \times _\alpha }{\mathbf{N}}$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1126190-X
Keywords: $ {C^{\ast}}$-algebra, endomorphism, covariant representation, crossed product, simple $ {C^{\ast}}$-algebra
Article copyright: © Copyright 1993 American Mathematical Society

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