A typical example where I know the $K$-theory of the quotient, and wonder if this could help to compute the $K$-theory of the extension. (Or other way round: knowing one part out of three ones.)
I have a certain exact sequence $$0 \rightarrow J \rightarrow F \rightarrow F/J \rightarrow 0$$ of $C^*$-algebras. I put it into the $K$-theory cyclic exact sequence and get $$ \begin{matrix} K_0(J) & \rightarrow & K_0(F) & \rightarrow & \bigoplus_{n=1}^\infty \mathbb{Z}\\ \uparrow & && & \downarrow^{\bf 0}\\ 0 & \leftarrow & K_1(F)& \leftarrow & K_1(J) \end{matrix} $$
Hence $$K_1(F) = K_1(J)$$ $$K_0(F)/K_0(J) = \bigoplus_{i=1}^\infty \mathbb{Z}$$
Any idea how to compute $K_i(F)$?
Here, $F$ is the universal $C^*$-algebra of the free inverse semigroup generated by $n$ generators $s_1, \ldots, s_n$, and $J$ is the two-sided closed ideal of $F$ generated by the relations $$s_i s_i^* s_j s_j^* \quad \forall 1 \le i \neq j \le n$$ (range projections of the generators are orthogonal).
Similarly one has $$ \begin{matrix} K_0(I) & \rightarrow & K_0(F) & \rightarrow & \mathbb{Z}_{n-1}\\ \uparrow & && & \downarrow^{\bf 0}\\ 0 & \leftarrow & K_1(F)& \leftarrow & K_1(J) \end{matrix} $$ because of the exact sequence $$0 \rightarrow I \rightarrow F \rightarrow \mathcal{O}_n \rightarrow 0$$ for the Cuntz algebra $\mathcal{O}_n$.
Does such information help for the computation of $F$ or is it unpromisingly if you know only one algebra out of three ones?
