## Examples of Idempotents

If $e$ is an idempotent in an algebra, so is $ueu^{-1}$ whenever $u$ is invertible. So, if $e$ and $f$ are homotopic in $M_\infty(A)$ if they are homtopic in some $M_n(A)$ where $A$ is a C*-algebra. Note that $\begin{pmatrix} I & \\ & 0\end{pmatrix}e_t \begin{pmatrix} I & \\ & 0\end{pmatrix}$ is a homotopy if $e_t:[0,1]\to M_\infty(A)$ is.

Examples of idempotents in $M_\infty(\mathbb{C})$. (a)\begin{pmatrix} & & & d\\ & & & c\\ & \Huge 0 & & b \\ & & & a \\ & & & 1\end{pmatrix} (b)$\frac{1}{1-t^2}\begin{pmatrix} -t^2 & t\\ -t &1\end{pmatrix}(t\in(1,\infty))$