I am studying the Elliot's Classification Theorem of AF-algebras (see Theorem 7.3.4 in An Introduction to $K$-theory for $C^*$-algebras by Rørdam, Larsen and Laustsen). I am trying to understand the hypothesis of the theorem. In particular, I wonder what is the role of $[1_A]_0$ and $[1_B]_0$ in the theorem. In other words, I am looking for an example of two unital AF-algebras $A$ and $B$ such that $K_0(A)\cong K_0(B)$ and $K_0(A)^+\cong K_0(B)^+$, but $A\not\cong B$.
Thanks everybody for your help.
I'm not that much into $K$-theory, but how about this (correct me if I'm wrong):
Let $A=\mathbb{C}$ and $B=M_2(\mathbb{C})$. They are clearly non-isomorphic, however $K_0(A)\cong K_0(B)$ as ordered abelian groups: consider the $*$-homomorphism $\iota:A\to B$ given by $x\mapsto\begin{pmatrix}x&0\\0&0\end{pmatrix}$ and the induced map $K_0(\iota):K_0(A)\to K_0(B)$. This is an isomorphism of abelian groups (see proposition 4.3.8 in your reference) and it actually preserves the order structure: we have $K_0(A)^+=\{[p]_0:p\in P_\infty(A)\}$. Now if $p\in P_\infty(A)$ is a projection then $K_0(\iota)([p]_0)=[\iota(p)]_0$ where by $\iota$ here I mean the natural map induced $\iota:P_\infty(A)\to P_\infty(B)$. This shows that $K_0(\iota)(K_0(A)^+)\subset K_0(B)^+$. On the other hand, if $q\in P_\infty(B)$ is a projection, then there exists some $n$ so that $q\in M_n(M_2(\mathbb{C}))\cong M_{2n}(\mathbb{C})$. Now $q$ has a certain rank, say $0\le d\le 2n$ so $[q]_0$ is equal to $d[q_1]_0$, where $q_1$ is the projection of $M_{2n}(\mathbb{C})$ with $1$ in the $(1,1)$ slot and zero elsewhere. But clearly $[q_1]_0=K_0(\iota)[p_1]_0$, where $p_1\in M_n(\mathbb{C})$ is the projection with $1$ in the $(1,1)$ slot and zero elsewhere and thus $[q]_0=d[q_1]_0=dK_0(\iota)([p_1]_0)=K_0(\iota)(d[p_1]_0)$. But now $d[p_1]_0=[p]_0$, where $p\in M_{2n}$ is any rank $d$ projection. This shows that $K_0(B)^+\subset K_0(\iota)(K_0(A)^+)$.