I have two questions with a relation to Elliot's classification of Approximately Finite $C*$-algebras (from now on referred to as AF-algebras).
Elliot's classical result yields that whenever $A$ and $B$ are AF-algebras and $\sigma \colon K_0(A) \to K_0(B)$ is a an isomorphism of scaled ordered $K_0$-groups, then this isomorphism induces a $*$-ismorphism $\varphi \colon A \to B$. Furthermore, it holds that $K_0(\varphi) = \sigma$. A proof of this is to be found in Blackadar's "$K_0$-theory for operator algebras".
I have been told that the above result is in fact slightly stronger. Namely that if $\tau \colon A \to B$ is another $*$-isomorphism then $K_0(\varphi) = K_0(\tau) = \sigma$, meaning that the lift is in fact uniquely determined. How does one obtain this fact?
My second question is of a slightly different kind. I have found a result saying that whenever one have $*$-homomorphisms $\iota_1, \iota_2 \colon C \to D$ where $C$ and $D$ are finite-dimensional $C*$-algebras, then $\iota_1$ are $\iota_2$ are unitarily equivalent. Again, I have been told that this may be generalised in the following sense: If we instead suppose for $C$ and $D$ to be AF-algebras, then $\iota_1$ and $\iota_2$ are approximately unitarily equivalent. It is my impression that Wikipedia mentions this result here, but I cannot find any reference for it. Does anyone have a reference or an idea as to how one obtains the result?
In your second question, I guess you mean $\iota_1,\iota_2 \colon C \to D$ are such that $K_0(\iota_1) = K_0(\iota_2)$.
Note that the following is true (Lemma 7.3.2. in "An Introducion to K-theory for C*-algebras" by Rordam, Larsen, Lausten)
Lemma. Assume $A$ is finite dimensional and $B$ is a unital C*-algebra with the cancellation property (for example an AF-algebra). Then the following holds: If $\varphi, \psi \colon A \to B$ are $*$-hom., then $K_0(\varphi) = K_0(\psi)$ if and only if there exists a unitary $u \in B$ such that $\varphi = u^* \psi(-) u$.
This lemma immediately answers your second question.
It also answers your first question: Whenever $A,B$ are AF algebras and $\varphi,\psi \colon A \to B$ induce the same map on $K$-theory, $\varphi$ and $\psi$ are approximately unitarily equivalent.