# Another way for defining $K_1$ group for a C*-algebra

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:

• It seems when we define $K_0$ like above, we add some 0s to the matrices and when we calculate the $Ker$, we nicely remove those 0s. – Peg Leg Jonathan Jul 11 '20 at 7:41
• anyone can help? – Peg Leg Jonathan Jul 11 '20 at 10:16
• This is how $K_1$ is already defined. Since $K_1(\mathbb C) = 0$ you have $\ker (K_1(A^+) \to K_1(\mathbb C)) = K_1(A^+) = K_1(A)$. – Jamie Gabe Jul 11 '20 at 10:51