Thank you for answering my question.

I have another question about the $K_1$ group. As you may know, some books define the $K_1$ group like below:

Also, it defines the $K_0$ group for an arbitrary C*-algebra like below:

Where $K_{00}(A)$ is the Grothendieck group for the abelian semigroup $V(A)$.

My question is how can I define $K_1(A)$ like the way of $K_0(A)$ more precisely how can I define $K_1(A)$ like below: