Elliott's program for nuclear C*-algebras deals with the problem of classifying nuclear C*-algebras by K-theoretical invariants. A major open question in this context is the UCT problem.

A separable C*-algebra $A$ is said to satisfy the UCT if for every separable C*-algebra $B$ a short exact sequence of the form

$0\rightarrow\text{Ext}\left(K_{0}\left(A\right)\text{, }K_{0}\left(B\right)\right)\rightarrow KK\left(A\text{, }B\right)\rightarrow\text{Hom}\left(K_{0}\left(A\right)\text{ }K_{0}\left(B\right)\right)\rightarrow0$

exists, where the right hand map is the natural one and the left hand map is the inverse of a map that is always defined. The UCT problem states that every separable, nuclear C*-algebra satisfies the UCT.

I'm not sure yet why the UCT is of such great importance concerning Elliott's program since I don't see how it's correctness provides a classification like for example in the AF-algebraic case. Can someone explain? Further I'm interested in the history of Elliott's program and how it led to the UCT question. Are there any good sources about the history of this topic?