Dilations of C*-Correspondences and the Simplicity of Cuntz–Pimsner Algebras

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.