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Exel's crossed product and relative Cuntz–Pimsner algebras

Published online by Cambridge University Press:  01 December 2006

NATHAN BROWNLOWE  and
IAIN RAEBURN
NATHAN BROWNLOWE
Affiliation:
School of Mathematical and Physical Sciences, University of Newcastle, NSW 2308, Australia. e-mail: [email protected]
IAIN RAEBURN
Affiliation:
School of Mathematical and Physical Sciences, University of Newcastle, NSW 2308, Australia. e-mail: [email protected]
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Abstract

We consider Exel's new construction of a crossed product of a $C^*$-algebra $A$ by an endomorphism $\alpha$. We prove that this crossed product is universal for an appropriate family of covariant representations, and we show that it can be realised as a relative Cuntz–Pimsner algbera. We describe a necessary and sufficient condition for the canonical map from $A$ into the crossed product to be injective, and present several examples to demonstrate the scope of this result. We also prove a gauge-invariant uniqueness theorem for the crossed product.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 141 , Issue 3 , November 2006 , pp. 497 - 508
Copyright
© 2006 Cambridge Philosophical Society

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