= Coleman trace operator
{c}
{title2=$\varphi\psi(f)=p^{-1}\sum_{\xi^p=1}f(\xi(1+T)-1)$}
In the unique decomposition $f=\sum_{i=0}^{p-1}(1+T)^i\varphi(f_i)$, define $\psi(f)=f_0$. Averaging translates over the p-th <roots of unity> annihilates every nonzero basis index and retains the zeroth, proving the displayed identity and its integrality. Thus $\psi\varphi=1$ and $\psi(\varphi(g)f)=g\psi(f)$. Under the <Amice transform>, $\psi=0$ characterizes measures supported on $\mathbb Z_p^\times$.
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