Let $A$ be a unital $C^*$-algebra and $a\in A$. If $aa^*+1$ is invertible in $A$ then the element $$\beta(a)=(aa^*+I)^{-1}\left(\begin{array}{cc}aa^* & a \\a^* & I\end{array}\right)$$ is an idempotent. The formula is similar to the one defining the Bott projection in the classical case.
I wanted to know what does the above $\beta$ mean for the $K$-theory of $A$? Does it have any significance similar to the classical Bott projection?