TY - JOUR
AU - Woodcock, C. F.
AB - A SPECTRAL VALUATION ON THE RING OF p-ADIC INTEGERS C. F. WOODCOCK 1. Introduction Throughout, Z , Q and C will respectively denote the ring of p-adic integers, p p p the field of p-adic numbers and the completion of the algebraic closure of Q (p prime). Let v denote the p-adic valuation of C normalised so that v (p) = 1. p p p In §2 we outline some basic results concerning the 'integral' 7 and the associated convolution multiplication * on W3)(Z ), the space of all uniformly differentiable functions / : Z ->• C (see [11]). p p In §3 and §4 we develop the 'Gelfand theory' for tfl3){Z ), * explicitly, showing in particular that, unlike / , the associated spectral valuation W is translation invariant. In §5 we prove that if L is a closed subfield of C then W is equivalent to the original valuation V on °U<2){Z , L) if and only if L has a 'non-zero different over Q p p (see [4]). Finally, in §6 we show that there is a convolution multiplication *j similarly associated to each 'integral' J on °U2)(Z ) but that, up to dilation, 7 is 
TI - A Spectral Valuation on the Ring of p‐Adic Integers
JF - Journal of the London Mathematical Society
DO - 10.1112/jlms/s2-25.2.223
DA - 1982-04-01
UR - https://www.deepdyve.com/lp/wiley/a-spectral-valuation-on-the-ring-of-p-adic-integers-YdHndKBnzO
SP - 223
EP - 234
VL - s2-25
IS - 2
DP - DeepDyve
ER -