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Jun 6, 2022 at 16:08 comment added Caleb Eckhardt Exercise 19.9.6 of Blackadar's text K-theory for Operator algebras seems to have what you are looking for when Y is a smooth manifold. Bruce cites the 1984 paper of Connes and Skandalis for that exercise.
Jun 6, 2022 at 11:32 history asked Jake Wetlock CC BY-SA 4.0