Dilations of C*-Correspondences and the Simplicity of Cuntz–Pimsner Algebras |
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Authors: | Jürgen Schweizer |
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Institution: | Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076, Tübingen, Germanyf1 |
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Abstract: | We develop a dilation theory for C*-correspondences, showing that every C*-correspondence E over a C*-algebra A can be universally embedded into a Hilbert C*-bimodule XE over a C*-algebra AE such that the crossed product A E
is naturally isomorphic to AE XE
. The Cuntz–Pimsner algebra
E is isomorphic to
E![times sign, right closed times sign, right closed](http://www.sciencedirect.com/scidirimg/entities/22ca.gif)
E
where
E and
E are quotients of AE, resp. XE. If E is full and the left action is by generalized compact operators, then
E is an equivalence bimodule or, equivalently, an invertible C*-correspondence. In general,
E is merely an essential Hilbert C*-bimodule. Slightly extending previous results on crossed products by equivalence bimodules, we apply our dilation theory to show that for full C*-correspondences over unital C*-algebras,
E is simple if and only if E is minimal and nonperiodic, extending and simplifying results of Muhly and Solel and Kajiwara, Pinzari, and Watatani. |
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Keywords: | |
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