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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?

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  • $\begingroup$ Probably nothing. It is unitarily conjugate to the projection $1 \oplus 0$ (as is evident by looking at the row $(a, 1)$, and for its complement), and of course there is a path of projections to the standard one from it. $\endgroup$ Jul 28, 2016 at 19:50
  • $\begingroup$ How about when viewed as a function of $a$? $\endgroup$
    – user95598
    Jul 30, 2016 at 16:14

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