Consider the following element:
$P_{n}=\sqrt{\frac{n}{π}}\int_{-\infty}^{+\infty}e^{-nt^{2}}τ_{t}(P)dt.$
where $P$ is a projection and $τ_{t}(P)=U(t)PU(t)^*$ for some group of unitaries in a von Neumann algebra.
Is $P_{n}$ a projection? How can we show that?
If you think of the discrete analogue of your formula, what you have is a convex combination of projections; this is usually not a projection.
With the above in mind, let $P=\begin{bmatrix}1&0\\0&0\end{bmatrix}$ and $V=\begin{bmatrix} 0&1\\1&0\end{bmatrix}$. Then $VPV^*=1-P$.
Let $s>0$ such that $\sqrt{\frac n \pi}\int_{-s}^s e^{-nt^2}\,dt=\frac12$. Define $$ U(t)=\begin{cases}1,&\ -s\leq t\leq s \\ V,&\ |t|>s\end{cases} $$ Then $P_n=\tfrac12\,P+\tfrac12(1-P)=\frac12$ is not a projection.
