The KK-lifting problem for dimension drop interval algebras

In this paper, we investigate the KK-theory of (generalized) dimension drop interval algebras (with possibly different dimension drops at the endpoints), with special emphasis on the problem which KK-class is representable by a ⁎-homomorphism between two such C*-algebras (allowing the tensor product with a matrix algebra for the codomain algebra). This lifting problem makes sense in its own right in KK-theory, and also has application to the classification of C*-algebras which are inductive limits of these building blocks. It turns out that when the dimension drops at the two endpoints are different, there exist KK-elements which preserve the order structure defined by M. Dadarlat and T.A. Loring in [4] on the mod p K-theory, but cannot be lifted to a ⁎-homomorphism. This is different from the equal dimension drop case, as shown by S. Eilers in [6].