Suppose that $A$ is a C*-algebra whose unitary group is contractible (e.g. $B(H)$ or more generally the stable multiplier algebra of any C*-algebra). It is clear from the definition that $K_1(A) = 0$, but I think it is also true that $K_0(A)$ is necessarily $0$. Is this correct? Is there an elementary proof?
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Here is an idea: To show that $K_1(A)=0$ you actually only use that $U(A)$ is path connected. Now it suffices to show that $U(\Sigma A)$ (unitary group of suspension) is path connected if $U(A)$ is contractible.