Dilations of C*-Correspondences and the Simplicity of Cuntz–Pimsner Algebras☆
Dilations of C*-Correspondences and the Simplicity of Cuntz–Pimsner Algebras☆
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C*-对应的膨胀和 Cuntz-Pimsner 代数的简单性☆
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发表时间：
2001
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通讯作者：
J. Schweizer
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作者：
J. Schweizer
我们发展了C*-对应的膨胀理论，证明C*-代数a上的每个C*-对应E可以普遍嵌入到C*-代数AE上的希尔伯特C*-双模XE中，使得交叉积a * * E N与AE * * XE Z自然同构。Cuntz-Pimsner代数OE与AE * * XE Z同构，其中AE和XE分别是AE的商。XE。如果E是满的，并且左作用是由广义紧算子完成的，则XE是一个等价双模，或者等价地是一个可逆C*对应。一般来说，XE仅仅是一个本质的Hilbert C*双模。我们推广了以前关于等价双模交叉积的结果，应用膨胀理论证明了对于单位C*-代数上的满C*-对应，当且仅当E是极小且非周期的，OE是简单的，扩展并简化了Muhly和Solel、Kajiwara、Pinzari和Watatani的结果。
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 N is naturally isomorphic to AE⋊XE Z. The Cuntz–Pimsner algebra OE is isomorphic to AE⋊XE Z where AE and XE are quotients of AE, resp. XE. If E is full and the left action is by generalized compact operators, then XE is an equivalence bimodule or, equivalently, an invertible C*-correspondence. In general, XE 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, OE is simple if and only if E is minimal and nonperiodic, extending and simplifying results of Muhly and Solel and Kajiwara, Pinzari, and Watatani.