Embedding C*-algebras into the Calkin algebra
In this talk I will present an application of forcing techniques to C*-algebras, objects coming from functional analysis. A C*-algebra is a norm-closed, self-adjoint subalgebra of B(H), the algebra of all linear bounded operators on a complex Hilbert space H. The Calkin algebra Q(H), defined as the quotient of B(H) modulo the ideal of compact operators, is a C*-algebra which, for many good reasons, is considered the noncommutative analogue of the boolean algebra P(omega)/Fin. In this talk I will prove that, given any C*-algebra A, there is a ccc forcing E_A which forces the existence of an embedding of A inside Q(H). The benchmark for our construction is the analogous fact holding for boolean algebras: given any boolean algebra B, there is a ccc forcing E_B which forces the existence of an embedding of B inside P(omega)/Fin. While the definition of E_B is elementary, its adaptation to C*-algebras is fairly involved and it requires deep results from operator algebras.