It suffices to check \Zp-equivariance of each map in the composition defining \Col, depending on Definition 9.5.1. The point is that the binomial series converges precisely on \sU_{\infty,1}. Writing f_u=\sum_{k\ge 0}a_k(u)T^k for the Coleman power series of u, one checks a_0(u)\equiv 1 \pmod{p}: indeed f_u(\pi_n)=u_n\equiv 1 \pmod{\pri_n} and, \pi_n being a uniformiser, this forces a_0(u)\equiv 1 \pmod{\pri_n}; as a_0(u)\in\Zp we get the congruence. Hence f_u-1\in(p,T), and by (p,T)-adic completeness of \Zp[[T]] the series f_u^a=\sum_{j\ge 0}\binom{a}{j}(f_u-1)^j converges. Since f_u(\pi_n)^a=u_n^a, the uniqueness of Coleman power series Theorem 9.2.2 gives f_u^a=f_{u^a}, so u\mapsto f_u is \Zp-equivariant. The logarithmic derivative satisfies \dlog(f_u^a)=a\,\dlog(f_u), hence is equivariant for the natural \Zp-action on \Zp[[T]]; and 1-\varphi\circ\psi, \partial^{-1} and \sA^{-1} are \Zp-linear by definition. Composing, \Col is \Zp-equivariant on \sU_{\infty,1}.

At each finite level n, since p is totally ramified in K_n there is a unique prime \pri_n above p, and reduction modulo \pri_n gives a short exact sequence 1\to\sU_{n,1}\to\sU_n\to\mu_{p-1}\to 1. The Teichmüller lift splits it, so \sU_n=\mu_{p-1}\times\sU_{n,1}. Passing to the inverse limit over n gives the claim.

A root of unity v\in\mu_{p-1}\subset\Zpx, viewed as the constant tower (v)_n, has constant Coleman power series f_v(T)=v. Constant series are killed by the logarithmic derivative \dlog, which differentiates. Hence v maps to 0 under \Col. Using Lemma 11.1.2, this shows the restriction to \sU_{\infty,1} loses nothing.

Since \dlog f=(1+T)f'/f, we have \dlog f=0 iff f'=0 iff f is a constant c. For a constant, the defining relation of the norm operator, \varphi(\cN f)=\prod_{\eta\in\mu_p}f((1+T)\eta-1), reads \varphi(\cN c)=c^p; as \varphi is the identity on constants this gives \cN(c)=c^p. Hence \cN-invariance forces c^p=c, and since c is a unit this means c\in\mu_{p-1}. Conversely every c\in\mu_{p-1} is a constant, lies in \WW, and is killed by \dlog. Therefore \ker(\dlog|_\WW)=\mu_{p-1}.

One checks equivariance map-by-map, using that \sigma_a acts on a power series by f(T)\mapsto f((1+T)^a-1). For u\mapsto f_u: evaluating (\sigma_a f_u)(\pi_n)=f_u((1+\pi_n)^a-1)=f_u(\xi_{p^n}^a-1)=\sigma_a(u_n), so it is equivariant. The logarithmic derivative satisfies the twisted relation \dlog(\sigma_a f)=a\,\sigma_a(\dlog f). On measures, restriction to \Zpx commutes with \sigma_a since multiplying the variable by a\in\Zpx stabilises both \Zpx and p\Zp. The operator \partial^{-1} obeys \partial^{-1}\circ\sigma_a=a^{-1}\sigma_a\circ\partial^{-1}, checked on measures via \int_{\Zpx} f\,\partial^{-1}\sigma_a\mu=\int_{\Zpx}\tfrac{f(ax)}{ax}\mu. Finally \sA^{-1} is equivariant by definition. The factor a from \dlog cancels the factor a^{-1} from \partial^{-1}, leaving \Col equivariant. This depends on Definition 9.5.1.

Combine Proposition 11.1.1 (\Zp-equivariance on \sU_{\infty,1}) with Proposition 11.1.5 (\GGam-equivariance). The two actions commute and together generate the \Lam(\GGam)-action, so \Col is \Lam(\GGam)-linear.