© 1997 by London Mathematical Society
© The London Mathematical Society
The Automorphism Group of a p-Adic Convolution Algebra
Institute of Mathematics and Statistics, The University Canterbury, Kent CT2 7NF. E-mail: C.F.Woodcock{at}ukc.ac.uk
Received 20 September 1994.
Throughout Zp and Qp will, respectively, denote the ring of p-adic integers and the field of p-adic numbers (for p prime). We denote by Cp the completion of the algebraic closure of Qp with respect to the p-adic metric. Let vp denote the p-adic valuation of Cp normalised so that vp(p)=1. Put 
for some n
0} so that Tp is the union of cyclic (multiplicative) groups 
of order pn (for n0).
Let UD(Zp) denote the Cp-algebra of all uniformly differentiable functions f:Zp
Cp under pointwise addition and convolution multiplication *, where for f, g
UD(Zp) and zZp we have
 |
the summation being restricted to i, j with vp(i+j–z)n.
This situation is a starting point for p-adic Fourier analysis on Zp, the analogy with the classical (complex) theory being substantially complicated by the absence of a p-adic valued Haar measure on Zp (see [5, 6] for further details).