Role of the UCT problem in classification theory for C*-algebras

Question

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?

Answer 1

Remark: The UCT-sequence is not correct as you stated it, it does not only involve the $K_0$-groups.

Regarding your second quastion: I highly recommend the book "Classification of nuclear $C^*$-algebras. Entropy in operator algebras" written by Rørdam and Størmer. You can find an explanation of the Elliott-conjecture which led to Elliott's classification program, which is as well outlined in this book.

The Eliott-conjecture asserts that all nuclear, separable $C^*$-algebras are classified up to the Elliott-invariant, which is a certain functor consiting of K-theory, traces and a pairing between traces and K-theory. The conjecture is known to be false in this generality, but true for certain subclasses of nuclear, separable $C^*$-algebras. Many of these subclasses have in common, that one assumpions for the $C^*$-algebras in these special subclasses is that they should satisfy the UCT. Thus, if you solve the UCT problem, then you can drop the UCT-assumption for these subclasses where the Elliott-conjecture is known to be true, and if the answer of this UCT-problem is yes, then these subclasses are fully understood. In short: The UCT is such a prominent ingredient for many classification theorems in the Elliott program, therefore one whishes to drop the UCT-assumption. But to drop the UCT-assumption, the UCT-problem should have a positive answer.

Regarding the first question (although I can not give a full answer of it right now), one possible situation (for Kirchberg algebras, for instance) is the following: For $C^*$-algebras satisfying the UCT, the sequence gives you a surjective map $$\gamma: \operatorname{KK}_*(A,B)\to \operatorname{Hom}(K_*(A),K_*(B)).$$ I.e. if the UCT is satisfied, for an isomorphism $f_*:K_*(A)\to K_*(B)$ in $\operatorname{Hom}(K_*(A),K_*(B))$ there exists a KK-equivalence $x\in \operatorname{KK}_*(A,B)$ such that $\gamma(x)=f_*$, i.e. $A$ and $B$ are KK-euqivalent. Sometimes it is possible to represent every element in $\operatorname{KK}_*(A,B)$ via a *-homomorphism $A\to B$ (see Phillip's classification theorem, theorem 8.2.1 in the recommended book, for instance) so that you can try to prove the existence of an isomorphsm $A\to B$ from there.