Using a group with Kazhdan’s property (T), in 1991 S. Wassermann constructed a separable C*-algebra A whose Ext is not a group. I will show that there is a unital embedding of A into the Calkin algebra such that the relative commutant of its range does not even include a unital copy of a simple, nonabelian C*-algebra (a moment of though shows that this implies Ext(A) is not a group).
The motivation for this result comes from the (widely open) question whether the Calkin algebra has a K-theory reversing automorphism. It implies that the Calkin algebra is not isomorphic to the corona of the stabilization of the Cuntz algebra O_\infty, with which it shares many properties (both algebras are purely infinite and simple, satisfy the double commutant theorem for separable unital subalgebras, have the same K-theory, and similarly behaved Ext functors).
The proof does not involve logic at all.
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