Show that $\mathbf{D}(\mathbb{C}\bigoplus \mathbb{C})$ is isomorphic to the additive semigroup $\mathbb{Z}^+\bigoplus \mathbb{Z}^+.$

Question

I am reading "An introduction to $C^*$ Algebra" by Rordam.

Show that $\mathbf{D}(\mathbb{C}\bigoplus \mathbb{C})$ is isomorphic to the additive semigroup $\mathbb{Z}^+\bigoplus \mathbb{Z}^+.$ I don't really get how this problem proceed but I know $D(\mathbb{C})\cong \{0,1,2,,...\} =\mathbb{Z^{+}}，$ where $\mathbb{Z}^+$ is equipped with the usual addition.

Anyone can give me some help?

Answer 1

Here is an idea for how to get started:

Recall how one shows that $\mathcal D(\mathbb C)\cong\mathbb Z^+$: Define a map $\mathcal P_\infty(\mathbb C)\to\mathbb Z^+$ by sending $p\in\mathcal P_n(\mathbb C)$ to $tr(p)$, the trace of $p$. Then one shows that this map descends to an additive map $\mathcal D(\mathcal C)\to\mathbb Z^+$, which is an isomorphism of semigroups.

Now note that if $p\in\mathcal P_n(\mathbb C\oplus\mathbb C)$, then we can write $p=(p_1,p_2)$, where $p_1,p_2\in\mathcal P_n(\mathbb C)$. Now define a map $\mathcal P_\infty(\mathbb C\oplus\mathbb C)\to\mathbb Z^+\oplus\mathbb Z^+$ by sending $p=(p_1,p_2)\in\mathcal P_n(\mathbb C\oplus\mathbb C)$ to $(tr(p_1),tr(p_2))$. Then show that this descends to an additive map $\mathcal D(\mathbb C\oplus\mathbb C)\to\mathbb Z^+\oplus\mathbb Z^+$, which is an isomorphism of semigroups.