Reference request: isomorphism in $K$-theory induced by inclusion of a full corner of a $C^*$-algebra

Question

Let $A$ be a $C^*$-algebra $p$ a projection such that $ApA$ is dense in $A$. Let $B=pAp$. Then it is known that the inclusion $$i\colon B\hookrightarrow A$$ induces an isomorphism on operator $K$-theory, i.e. the functorially induced map $$i_*\colon K_*(B)\to K_*(A)$$ is an isomorphism.

I wonder what the basic ideas are that underlie the proof. Is there a good reference for this?

Answer 1

This is Proposition 2.7.19 in Willett and Yu's book "Higher Index Theory" (you can find a copy of a draft of the book here, or from Rufus Willett's homepage). The idea of the proof is to reduce to the separable case, and appeal to the following result of Brown:

If $A$ is separable and $p\in M(A)$ is a full projection, then there is an isometry $v\in M(A\otimes\mathbb K)$ such that $vv^*=p\otimes 1$.

Using this, we define a map $\phi:A\otimes\mathbb K\to pAp\otimes\mathbb K$ by $\phi(a)=vav^*$. This is a $*$-isomorphism, and its composition with the inclusion $pAp\otimes\mathbb K\hookrightarrow A\otimes\mathbb K$ yields an isomorphism on $K$-theory, so in particular the inclusion (after tensoring with $\mathbb K$) induces an isomorphism on $K$-theory. Finally, we have a commutative diagram \begin{align*} \begin{array}{ccc} pAp & \hookrightarrow & A \\ \downarrow & & \downarrow \\ pAp\otimes\mathbb K & \hookrightarrow & A\otimes\mathbb K \end{array} \end{align*} where the vertical arrows are "upper left corner" inclusions. The vertical maps and the bottom map induce isomorphisms on $K$-theory, hence so does the top map.