Algebraic K-theory of stable C*-algebras

Let A be a unital C*-algebra and let l denote the Calkin algebra (the bounded operators on a separable Hilbert space, modulo the compact operators K). We prove the following conjecture of M. Karoubi: the algebraic and topological K-theory groups of the tensor product C*-algebra A ⊗ l are equal. The algebra A ⊗ l may be regarded as a "suspension" of the more elementary C*-algebra A ⊗ K; thus Karoubi's conjecture asserts, roughly speaking, that the algebraic and topological K-theories of stable C*-algebras agree.