The Automorphism Group of a p ‐Adic Convolution Algebra

Throughout Z p and Q p will, respectively, denote the ring of p ‐adic integers and the field of p ‐adic numbers (for p prime). We denote by C p the completion of the algebraic closure of Q p with respect to the p ‐adic metric. Let v p denote the p ‐adic valuation of C p normalised so that v p ( p )=1. PutT p = { ω ∈ C p | ω p n= 1for some n ⩾0} so that T p is the union of cyclic (multiplicative) groupsC p nof order p n (for n ⩾0). Let UD(Z p ) denote the C p ‐algebra of all uniformly differentiable functions f :Z p →C p under pointwise addition and convolution multiplication *, where for f, g ∈UD(Z p ) and z ∈Z p we have f * g ( z ) = lim n → ∞( ∑ i , j = 0p n − 1f ( i ) g ( j ) ) / p n ,the summation being restricted to i, j with v p ( i + j − z )⩾ n . This situation is a starting point for p ‐adic Fourier analysis on Z p , the analogy with the classical (complex) theory being substantially complicated by the absence of a p ‐adic valued Haar measure on Z p (see [ 5 , 6 ] for further details).