This is from the book "K-Theory and C* Algebra" by Wegge Olsen.
I would like to understand the summation of extensions as it described there, but I am having some trouble. I don't see how Proposition 3.3.4. helps with showing Corollary 3.3.5.
Also what I don't get here is that $\tau_{1} \oplus \tau_{2}$ should lead to something in $\mathcal{M}(A)/A \oplus \mathcal{M}(A)/A$, but it leads us here to $\mathcal{M}(A)/A$ it seems hence there must be some embedding here, which I don't really see how to get. Remark 3.3.6. shows an embedding of $\mathcal{M}(A^s)/A^s$ into $\mathbb{M_{2}}(\mathcal{M}(A^s)/A^s)$, is this the embedding that makes it possible? But this already embeds $\mathcal{M}(A^s)/A^s$ which we get after $S_{\phi}$ anyway? I am kind of confused here with these arguments. Would appreciate it if someone could shed some light on this.
The series of mappings described in Remark 3.3.6 appears to have a typo. It should read \begin{align*} C\overset{\tau_1\oplus\tau_2}{\longrightarrow} \left(\mathcal{M}(A^s)/A^s\right)\oplus\left(\mathcal{M}(A^s)/A^s\right)\hookrightarrow \mathcal{M}_2\left(\mathcal{M}(A^s)/A^s\right)\overset{S_\phi}{\longrightarrow}\mathcal{M}(A^s)/A^s. \end{align*}