Shape theory and extensions of C*-algebras

Let A, A0 be separable C*-algebras, B a stable ¾-unital C*-algebra. Our main result is the construction of the pairing [[A';A]] £ Ext¡1=2(A;B) ! Ext¡1=2(A';B), where [[A';A]] denotes the set of homotopy classes of asymptotic homomorphisms from A' to A and Ext¡1=2(A;B) is the group of semi-invertible extensions of A by B. Assume that all extensions of A by B are semi-invertible. Then this pairing allows us to give a condition on A' that provides semi-invertibility of all extensions of A' by B. This holds, in particular, if A and A' are shape equivalent. A similar condition implies that if Ext¡1=2 coincides with E-theory (via the Connes-Higson map) for A then the same holds for A'