Product systems of C*-correspondences and Takai duality
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PRODUCT SYSTEMS OF C∗ -CORRESPONDENCES AND TAKAI DUALITY BY
Elias Katsoulis Department of Mathematics,East Carolina University Greenville, NC 27858-4353, USA e-mail: [email protected]
ABSTRACT
We establish the Hao–Ng isomorphism for generalized gauge actions of locally compact abelian groups on product systems over abelian lattice orders and we then use it to explore Takai duality in this context. As an application we generalize some recent work of Schafhauser.
1. Introduction If (A, G, α) is a C∗ -dynamical system over a locally compact abelian group G, then the crossed product C∗ -algebra A α G admits a natural action α of the This produces a new C∗ -dynamical system (A α G, G, α dual group G. ) and the classical Takesaki–Takai duality asserts that A ⊗ K(L2 (G, μ)), (A α G) α G where K(L2 (G, μ)) denotes the compact operators on L2 (G, μ), μ being the Haar measure on G. The Cuntz–Pimsner algebras of product systems of C∗ -correspondences over abelian lattice orders (G, P ), include as particular examples crossed products of C∗ -algebras by various discrete abelian groups. Just as in the case of a crossed r product, the generic Cuntz–Pimsner algebra N OX admits a natural action of Received May 26, 2019 and in revised form September 19, 2019
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the so called gauge action. One would like to calculate the the dual group G, r in particular one wonders what is the associated crossed product N OX G; analogue of the classic Takesaki–Takai duality in this case. This is the main theme of this paper with inspiration coming from the work of Abadie [1] who first studied this problem for Cuntz–Pimsner algebras of C∗ -correspondences. r involves the solution of the Hao–Ng A key step in the calculation of N OX G isomorphism problem for generalized gauge actions of locally compact abelian groups on product systems over abelian lattice orders. (See Section 3 for a statement of the Hao–Ng isomorphism problem and [20] for a detailed discussion of its impact on current operator algebra research.) Recent advances in the theory of non-selfadjoint operator algebras by Dor-On and the author [9] allow us to do this in Theorem 3.8. A key element in the proof of Theorem 3.8 is Proposition 3.2 which gives a very workable criterion for checking the compact alignment of a product system. Using the Hao–Ng isomorphism and the Fourier transform, we calculate the r by the gauge action. Indeed in Theorem 4.2 we crossed product N OX G r as the Cuntz–Pimsner algebra of a give a very concrete picture for N OX G product system that involves only the group G and not its dual. For product systems of regular and full C∗ -correspondences over abelian lattice orders we r is Morita equivalent can do more. In Corollary 4.7 we show that N OX G r to the core of N OX , thus offering a true generalization of the Takai duality in our context. As a consequence of Corollary 4.7, we are able to generalize some recent results of Schafhauser [38] in our context; this is discussed in Section 5.
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